Graphing Linear Functions — Free Algebra 1 Review Games.

In y = mx + b form, m is the slope. Here m = 3.

In y = mx + b, b is the y-intercept. Here b = 5.

Horizontal lines have no rise, so slope = 0.

Slope = (8-2)/(3-1) = 6/2 = 3.

Vertical lines have zero run, making the slope undefined.

Set y = 0: 2x = 6, x = 3.

Using y = mx + b with m = 2 and b = -1: y = 2x - 1.

Parallel lines have the same slope. The slope is -4.

Perpendicular slope is the negative reciprocal: -1/2.

Plug in: 3(2) + 1 = 7. Yes, the point satisfies the equation.

Rewrite: -4y = -3x + 12, y = (3/4)x - 3. Slope = 3/4.

The product of slopes is (2/3)(-3/2) = -1, so they are perpendicular.

Slope = (-7-5)/(4-(-2)) = -12/6 = -2.

Both points have y = 4, so it is a horizontal line: y = 4.

Plug into y = mx + b: y = (-1/2)x + 3.

In slope-intercept form y = mx + b, m is the coefficient of x and represents the slope of the line — how steep it is and in which direction it tilts. The variable b represents the y-intercept. Confusing m and b is a very common mistake.

Slope-intercept form is y = mx + b, where the equation is solved for y and written as a slope times x plus a constant. Only y = -x + 5 fits this structure, with slope m = -1 and y-intercept b = 5. Choice A is standard form, choice B is point-slope form, and choice D is not solved for y.

In y = 5x there is no constant term, so b = 0. The line passes directly through the origin (0, 0), making the y-intercept 0. The value 5 is the slope, not the y-intercept. Any line written as y = mx with no added constant has a y-intercept of 0.

The slope is the coefficient of x. In y = 3x - 7, the coefficient is 3, which is positive. Choices A and B have negative slopes (-2 and -1). Choice C, y = 4, is a horizontal line with a slope of zero — not a positive slope.

Using slope-intercept form y = mx + b with m = -5 and b = 2 gives y = -5x + 2. Choice A incorrectly swaps the slope and y-intercept. Choice C uses the correct slope but makes the y-intercept negative, which contradicts the given value of 2.

To find the x-intercept, set y = 0 and solve: 0 = 4x - 8, so 4x = 8 and x = 2. The x-intercept is 2. Choice A (-8) is the y-intercept — the value you get when you set x = 0 instead of y = 0, which is a common mix-up.

A line sloping downward from left to right has a negative slope because as x increases, y decreases — the rise is negative while the run is positive, so rise/run is negative. A positive slope rises from left to right, a zero slope is horizontal, and an undefined slope is vertical.

To find the y-intercept, set x = 0: 2(0) - 3y = 12, so -3y = 12 and y = -4. You can confirm by converting to slope-intercept form: y = (2/3)x - 4, where the y-intercept is clearly -4. The constant 12 in the original equation is not the y-intercept.

To find the x-intercept, set y = 0: 0 = -2x + 6, so 2x = 6 and x = 3. The x-intercept is 3. A common mistake is reporting 6, which is the y-intercept found by setting x = 0, not y = 0.

The slope of y = (3/4)x + 2 is 3/4. A perpendicular line has the negative reciprocal slope: flip 3/4 to get 4/3, then negate to get -4/3. Since the line passes through the origin, b = 0, giving y = -(4/3)x. Choice C has the right reciprocal but the wrong sign. Choice B keeps the original fraction and only changes the sign.

Parallel lines have equal slopes and different y-intercepts. In choice B, both lines have slope 2 but different y-intercepts (4 and -7), so they are parallel. Choice A has slopes 3 and -3, which are not equal. Choice C has slopes 5 and 1/5, which multiply to 1 — those lines intersect. Choice D has slopes -1 and 1.

The x-intercept -2 gives the point (-2, 0) and the y-intercept 4 gives (0, 4). Using the slope formula: m = (4 - 0) / (0 - (-2)) = 4/2 = 2. A common error is computing (0 - 4)/(0 - (-2)) = -4/2 = -2, which reverses the subtraction order in the numerator.

The y-intercept of y = 4x + 3 is 3. The slope of y = 2x - 1 is 2, so the perpendicular slope is -1/2 (negative reciprocal). Combining these: y = -(1/2)x + 3. Choice D has slope -2, which is just the negative of the original slope, not its negative reciprocal — a frequent error when finding perpendicular slopes.

The line through (0, -2) with slope 5 has equation y = 5x - 2. Substituting x = 3: y = 5(3) - 2 = 15 - 2 = 13. Choice C (7) comes from computing 5 + 2 instead of 5(3) - 2. Choice D (15) forgets to subtract the y-intercept after multiplying.

Solve for y: subtract 4x from both sides to get 2y = -4x + 10, then divide every term by 2 to get y = -2x + 5. The slope is -2 and the y-intercept is 5. Choice D only subtracts 4x without dividing the entire equation by 2, leaving coefficients that are twice the correct values.

Using the slope formula m = (y2 - y1)/(x2 - x1): m = (0 - 6)/(3 - 0) = -6/3 = -2. The slope is negative because the line falls from left to right. Choosing +2 is a sign error that comes from accidentally computing (6 - 0)/(3 - 0) instead of maintaining consistent subtraction order.

Substitute x = 3 and y = 11 into y = 2x + b: 11 = 2(3) + b, so 11 = 6 + b and b = 5. Choice D (17) results from adding rather than subtracting: 11 + 6 = 17 instead of correctly solving b = 11 - 6 = 5.

Slope of Line A: (6 - 2)/(4 - 0) = 4/4 = 1. Slope of Line B: (3 - (-1))/(2 - 0) = 4/2 = 2. The slopes are different, so the lines are not parallel. Their product is 1 x 2 = 2, which is not -1, so they are not perpendicular either. Lines with different slopes that are not negative reciprocals intersect at exactly one point.

The slope of y = 4x - 1 is 4. The perpendicular slope is -1/4 (negative reciprocal). Using point-slope form with (2, 3): y - 3 = -(1/4)(x - 2), so y = -(1/4)x + 1/2 + 3 = -(1/4)x + 7/2. Choice D uses slope -4, which is only the negative of 4 — not the negative reciprocal — and is a common mistake when identifying perpendicular slopes.

Find the slope: m = (-3 - 5)/(3 - (-1)) = -8/4 = -2. Then use point-slope form with (-1, 5): y - 5 = -2(x - (-1)) = -2(x + 1), so y = -2x - 2 + 5 = -2x + 3. Choice C has the correct slope but a sign error on the constant. Choice D results from a sign error when distributing: -2(x + 1) gives -2x - 2, not -2x + 2.

If the x-intercept is 6, then the point (6, 0) lies on the line. Substituting into ax + 3y = 12: a(6) + 3(0) = 12, so 6a = 12 and a = 2. Choice B (6) confuses the x-intercept value itself with the coefficient a. Choice C (3) is the coefficient already present in the equation.

Perpendicular lines have slopes whose product equals -1. For choice D: (2/3) x (-3/2) = -6/6 = -1. Those lines are perpendicular. For choice C: (3/4) x (4/3) = 1, not -1 — those lines intersect but are not perpendicular. Choices A and B each pair lines with equal slopes, making them parallel.

Convert L1 to slope-intercept form: -3y = -2x + 6, so y = (2/3)x - 2. The slope of L1 is 2/3. Parallel lines share the same slope, so L2 also has slope 2/3. Since L2 passes through (0, 4), its equation is y = (2/3)x + 4. Choice D has the correct slope but uses the y-intercept of L1 (-2) instead of the given y-intercept of L2 (4).

First find the intercepts. X-intercept: set y = 0 to get 0 = 2x - 6, so x = 3, giving the point (3, 0). Y-intercept: set x = 0 to get y = -6, giving (0, -6). The triangle has vertices at the origin (0, 0), (3, 0), and (0, -6), with base 3 and height 6. Area = (1/2)(3)(6) = 9. Choice D (18) results from forgetting to multiply by 1/2.

Using the slope formula: (7 - 3)/(5 - a) = 2, so 4/(5 - a) = 2. Multiply both sides by (5 - a): 4 = 2(5 - a) = 10 - 2a. Solving: 2a = 6 and a = 3. You can verify: slope = (7 - 3)/(5 - 3) = 4/2 = 2. Choice B (a = 1) gives slope 4/(5 - 1) = 1, not 2.

In slope-intercept form y = mx + b, m is the slope and b is the y-intercept. Here m = -5, so the slope is -5. The value 3 is the y-intercept, not the slope.

In slope-intercept form y = mx + b, b is the y-intercept. Here b = -7. The value 4 is the slope. A common error is confusing the slope with the y-intercept.

A horizontal line has no rise (the change in y is 0), so slope = 0 divided by run = 0. A vertical line, by contrast, has undefined slope because its run is 0 and division by zero is undefined.

A vertical line has no run (change in x = 0), so the slope formula produces division by zero, which is undefined. This is different from a horizontal line, which has slope 0.

To find the x-intercept, set y = 0 and solve: 0 = 3x - 6, so 3x = 6 and x = 2. The x-intercept is (2, 0). The point (0, -6) is the y-intercept, not the x-intercept.

In y = mx + b, the coefficient m is the slope. Only y = 3x - 4 has a positive slope (m = 3). The other choices have slopes of -2, -1, and -4, all of which are negative.

Slope-intercept form is y = mx + b, where the equation is solved for y with slope and y-intercept clearly visible. Choices A and D are in standard form, and Choice B is in point-slope form. Only Choice C matches y = mx + b directly.

The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0. The x-intercept is where the line crosses the x-axis. Lines extend infinitely in both directions, so they have no highest point or midpoint.

Using the slope formula: m = (y2 - y1)/(x2 - x1) = (5 - 1)/(3 - (-3)) = 4/6 = 2/3. A common error is inverting the fraction to get 6/4 = 3/2, which results from swapping the numerator and denominator.

Substituting m = -3 and b = 5 into y = mx + b gives y = -3x + 5. Choice A swaps the slope and y-intercept values, and Choice C uses a positive slope instead of a negative one.

Solve for y: subtract 3x from both sides to get -y = -3x + 9, then multiply every term by -1 to get y = 3x - 9. A common mistake is flipping only one sign, producing y = -3x + 9 or y = 3x + 9.

Parallel lines have equal slopes. The given line has slope 2/3, so the parallel line must also have slope 2/3. Choice C has slope 2/3, so it is parallel. Choice A has slope 3/2, and Choice B has slope -3/2, which is the perpendicular slope.

Substitute the known values into y = mx + b: 1 = 4(2) + b, so 1 = 8 + b, and b = 1 - 8 = -7. Choice B (9) comes from adding 1 + 8 = 9 instead of isolating b by subtracting.

Steepness is measured by the absolute value of the slope. The given line has |slope| = 2. The absolute slopes of each choice are |1| = 1, |3/2| = 1.5, |-3| = 3, and |1/2| = 0.5. Only Choice C has a larger absolute slope value (3 > 2).

Perpendicular lines have slopes that are negative reciprocals of each other. The given slope is 1/4, so the perpendicular slope is -1 divided by (1/4) = -4. Choice B (4) takes the reciprocal but omits the required sign change.

Since the line passes through (0, 6), the y-intercept is b = 6. With slope m = -2, substituting into y = mx + b gives y = -2x + 6. Choice A incorrectly uses -6 as the y-intercept instead of 6.

Set x = 0: 5(0) + 2y = 10 → 2y = 10 → y = 5. The y-intercept is 5. Alternatively, solving for y gives y = -5/2 x + 5, confirming b = 5. Choice D (10) is the constant on the right-hand side of the equation, not the y-intercept.

Find the slope: m = (5 - (-1))/(-4 - 2) = 6/(-6) = -1. Then substitute point (2, -1): -1 = -1(2) + b → -1 = -2 + b → b = 1. The equation is y = -x + 1. Choice B incorrectly uses a positive slope of 1.

Rewrite the second line: 3y = -6x + 15 → y = -2x + 5, so its slope is -2. Rewrite the first line: ky = -2x + 8 → y = (-2/k)x + 8/k, so its slope is -2/k. Setting slopes equal: -2/k = -2 → k = 1. Choice C (k = 3) would produce slope -2/3, which does not match -2.

Find both intercepts: x-intercept (set y = 0): 0 = -2x + 8 → x = 4, giving point (4, 0). Y-intercept: (0, 8). The triangle has base 4 along the x-axis and height 8 along the y-axis. Area = (1/2)(4)(8) = 16. Choice A (8) is just the y-intercept value, not the area.

The perpendicular slope is the negative reciprocal of 3/2, which is -2/3. Using point (3, 1): 1 = (-2/3)(3) + b → 1 = -2 + b → b = 3. The equation is y = -(2/3)x + 3. Choice A uses the reciprocal 2/3 without the required negative sign.

For perpendicular lines, the product of their slopes equals -1. The second line has slope 1/2, so a times (1/2) = -1 → a = -2. Choice A (2) ignores the sign requirement; multiplying 2 by 1/2 gives 1, not -1.

Find the slope: m = (9 - 3)/(4 - 1) = 6/3 = 2. Use point (1, 3): 3 = 2(1) + b → b = 1. Equation: y = 2x + 1. Substitute y = 15: 15 = 2k + 1 → 2k = 14 → k = 7. Choice B (8) results from an arithmetic error when isolating k.

Line Q is perpendicular to P (slope -4), so Q's slope is the negative reciprocal: -1 divided by (-4) = 1/4. Line R is parallel to Q, so R has the same slope as Q, which is 1/4. Choice D (-1/4) is the negative reciprocal of 1/4, making it perpendicular to Q rather than parallel.

Find the slope: m = (9 - 5)/(6 - 2) = 4/4 = 1. Using point (2, 5): 5 = 1(2) + b → b = 3. Equation: y = x + 3. Set y = 0: 0 = x + 3 → x = -3. The x-intercept is (-3, 0). Choice B (3, 0) confuses the y-intercept value with the x-intercept coordinate.

In slope-intercept form y = mx + b, the coefficient m is the slope and b is the y-intercept. Here m = -5, so the slope is -5. The value 3 is the y-intercept, not the slope.

In y = mx + b, b is the y-intercept. Here b = -9, so the y-intercept is -9. The value 4 is the slope, not the y-intercept.

Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. The form ax + by = c is standard form, y - y1 = m(x - x1) is point-slope form, and x/a + y/b = 1 is intercept form.

A horizontal line has zero rise for any run, so slope = rise/run = 0/run = 0. A vertical line (not a horizontal line) has undefined slope because the run equals zero, making division by zero impossible.

The x-intercept is where the line crosses the x-axis, meaning y = 0. Setting y = 0: 0 = x - 5, so x = 5. The x-intercept is (5, 0). The point (0, -5) is the y-intercept, not the x-intercept.

Slope = (y2 - y1) / (x2 - x1) = (8 - 0) / (4 - 0) = 8/4 = 2. The value 4 is the x-coordinate of the second point, and 1/2 is the reciprocal of the slope, not the slope itself.

In y = mx + b, the slope is m, the coefficient of x. For y = -2x + 5, m = -2, which is negative. The other choices have slopes of 3, 1, and 1/3 respectively, all of which are positive.

Slope = (y2 - y1) / (x2 - x1) = (9 - 1) / (1 - (-3)) = 8 / 4 = 2. A common mistake is computing 1 - (-3) as 2 instead of 4. Since 1 - (-3) = 1 + 3 = 4, the correct slope is 2.

Solve for y: 4x - 2y = 8 becomes -2y = -4x + 8, then dividing every term by -2 gives y = 2x - 4. A common error is dividing only some terms by -2, which incorrectly produces y = -2x + 4 (choice A).

To find the x-intercept, set y = 0: 3x + 4(0) = 12 gives 3x = 12, so x = 4. The x-intercept is (4, 0). Setting x = 0 instead gives 4y = 12, so y = 3, which is the y-intercept (0, 3), not the x-intercept.

Perpendicular lines have slopes that are negative reciprocals. The slope of y = 3x - 2 is 3, so the perpendicular slope is -1/3. Choice D has slope -1/3. Choice C has slope +1/3 (missing the negative sign), and choice B has slope -3, which is the negative but not the reciprocal of 3.

Use y = mx + b with m = -2 and the point (3, 4): 4 = -2(3) + b gives 4 = -6 + b, so b = 10. A common error is treating -2 times 3 as +6 instead of -6, which would incorrectly give b = -2.

Rewrite in slope-intercept form: -3y = -6x + 9 gives y = 2x - 3, so the slope is 2. A parallel line has the same slope of 2. Choice A (-2) results from forgetting to flip the sign when dividing both sides by -3.

Slope = (1 - (-3)) / (4 - 0) = 4/4 = 1. The point (0, -3) is the y-intercept, so b = -3. The equation is y = x - 3. Choice C (slope 2) is wrong because plugging in x = 4 gives y = 2(4) - 3 = 5, not 1.

The two intercept points are (6, 0) and (0, -3). Slope = (0 - (-3)) / (6 - 0) = 3/6 = 1/2. Choice A (2) is the reciprocal of the correct answer, and choice D (-1/2) incorrectly applies a negative sign when both the rise and run are positive.

First find the slope: (-3 - 5) / (3 - (-1)) = -8/4 = -2. Then use y = mx + b with point (-1, 5): 5 = -2(-1) + b gives 5 = 2 + b, so b = 3. Choice C (-2) is the slope, not the y-intercept.

Set y = 0: 0 = (3/4)x - 6 gives (3/4)x = 6, then x = 6 times (4/3) = 8. The x-intercept is (8, 0). Choice B (6, 0) is a common error from treating 6 as the answer without multiplying by the reciprocal of 3/4.

The y-intercept of y = 5x - 2 is -2. A matching line needs b = -2 and a slope other than 5. Choice B, y = -3x - 2, has y-intercept -2 and slope -3. Choice A has y-intercept +2 (not -2), and choice C has slope 5 (the same slope, not different).

Parallel lines share the same slope. The slope of y = -2x + 5 is -2. Using point-slope form: y - 1 = -2(x - 3) gives y - 1 = -2x + 6, so y = -2x + 7. Choice B uses slope 1/2, which is the perpendicular slope. Choice C has the correct slope but does not pass through (3, 1) since -2(3) - 5 = -11, not 1.

Rewrite in slope-intercept form: -3y = -kx + 6 gives y = (k/3)x - 2. The slope is k/3. Setting k/3 = 4 gives k = 12. Choice C (4) confuses k with the slope directly, ignoring that k is divided by 3 in slope-intercept form. Choice B results from incorrectly handling the sign when dividing by -3.

Find both intercepts. X-intercept (set y = 0): 2x = 10, so x = 5, giving point (5, 0). Y-intercept (set x = 0): 5y = 10, so y = 2, giving point (0, 2). The triangle has base 5 along the x-axis and height 2 along the y-axis. Area = (1/2)(5)(2) = 5. Choice B (10) is the result of forgetting to multiply by 1/2.

Rewrite 2x - 4y = 8 as y = (1/2)x - 2, so its slope is 1/2. The perpendicular slope is -2. Using point-slope form: y - 3 = -2(x - (-2)) gives y - 3 = -2x - 4, so y = -2x - 1. Choice B uses the original slope 1/2 instead of the perpendicular slope. Choice D has the correct perpendicular slope but the wrong y-intercept.

Rewrite 3x - y = 7 as y = 3x - 7 (slope = 3). Rewrite x + 3y = 9 as y = -(1/3)x + 3 (slope = -1/3). Since 3 times (-1/3) = -1, the lines are perpendicular. Choice D is incorrect because perpendicular slopes must be negative reciprocals of each other, not merely equal in magnitude and opposite in sign.

Find the equation: y - 6 = -3(x - 1) gives y = -3x + 3 + 6 = -3x + 9. Set y = 0: -3x + 9 = 0 gives x = 3. The x-intercept is (3, 0). Choice D is the y-intercept (0, 9). Choice B confuses the value of the slope with the x-intercept.

Slope from A to B: (6 - 2) / (3 - 1) = 4/2 = 2. Slope from B to C: (10 - 6) / (5 - 3) = 4/2 = 2. Because the slopes are equal and the segments share point B, all three points lie on the line y = 2x, making them collinear. Choice C is false because both slopes equal 2. Choice A is false because collinear points cannot enclose a triangle.

In slope-intercept form y = mx + b, the coefficient of x is the slope. Here m = -7. The value 2 is the y-intercept, not the slope. Confusing the slope and y-intercept is a very common error.

In y = mx + b, b is the y-intercept. Here b = -9. The value 4 is the slope, which is a common distractor. Remember: m is slope and b is the y-intercept.

A horizontal line has the form y = c and has no vertical rise. Slope = rise/run = 0/run = 0. Vertical lines, not horizontal ones, have undefined slope because their run equals zero.

A vertical line has the form x = c. Moving along it, x never changes, so run = 0. Since we cannot divide by zero, the slope is undefined. Horizontal lines have slope 0, which is commonly confused with vertical.

Slope = rise / run = change in y / change in x = (y2 - y1) / (x2 - x1). Choice A inverts the fraction, placing x over y. Choice D swaps numerator and denominator, producing the reciprocal of the correct slope.

Slope-intercept form is y = mx + b, where y is isolated on the left with coefficient 1. Choice A is standard form. Choice B is point-slope form. Choice C has x isolated rather than y.

Slope = (10 - 2) / (4 - 0) = 8 / 4 = 2. Choice A (8) is only the numerator, the rise, before dividing by the run. Choice C (1/2) inverts the fraction, giving run over rise instead of rise over run.

In slope-intercept form y = mx + b, m is the slope (rate of change) and b is the y-intercept. Choice B (b) is the y-intercept, the point where the line crosses the y-axis, and is frequently confused with the slope.

Substitute m = -3 and b = 5 into y = mx + b to get y = -3x + 5. Choice A swaps the slope and y-intercept values. Choice B uses the correct slope but makes the y-intercept negative.

Isolate y: subtract 6x to get -2y = -6x + 4, then divide every term by -2 to get y = 3x - 2. Choice B results from a sign error when dividing by -2. Choice C forgets to divide by the coefficient of y.

To find the x-intercept, set y = 0: 0 = 3x - 9, so 3x = 9 and x = 3. The x-intercept is (3, 0). Choice A is the y-intercept, found by setting x = 0. Choice D (9, 0) results from forgetting to divide by 3.

Substitute x = 1: y = 2(1) - 5 = -3, confirming (1, -3) is on the line. For (2, 1): y = 2(2) - 5 = -1, not 1. For (0, 5): y = 2(0) - 5 = -5, not 5. For (3, 2): y = 2(3) - 5 = 1, not 2.

Use y = mx + b with m = 2 and point (3, 1): 1 = 2(3) + b gives 1 = 6 + b, so b = -5. Choice C (7) is a common error from adding instead of subtracting: 1 + 6 = 7. Choice A (5) results from a sign error on the final step.

Solve for y: 10y = -5x + 20, so y = -(1/2)x + 2. The slope is -1/2. Choice C (1/2) drops the negative sign. Choice A (5) uses the coefficient of x from the original equation without first isolating y.

Parallel lines have equal slopes and different y-intercepts. Both lines in choice B have slope 2 but y-intercepts of 3 and -7, making them parallel. In choice A slopes are 4 and -4. In choice C slopes are 1 and 3. In choice D slopes are 1/2 and 2.

The intercepts give two points: (4, 0) and (0, -8). Slope = (-8 - 0) / (0 - 4) = -8 / (-4) = 2. Choice A (-2) is a common error from computing y-intercept divided by x-intercept as -8/4 = -2, which ignores the proper slope formula.

Set x = 0: 3(0) - 4y = 12 gives -4y = 12, so y = -3. The y-intercept is (0, -3). Choice A (3) results from forgetting to apply the negative sign when dividing both sides by -4.

The perpendicular slope is the negative reciprocal of 2/3, which is -3/2. Using point (4, 1): 1 = -(3/2)(4) + b gives 1 = -6 + b, so b = 7. Equation: y = -(3/2)x + 7. Choice D takes the reciprocal without negating it. Choice A is simply the original line.

A system has no solution when the lines are parallel, meaning equal slopes and different y-intercepts. The first line has slope 3, so k must equal 3. Since the y-intercepts are 2 and -4 (different), the lines are parallel and never intersect. Choice A (-3) gives lines with opposite slopes that do intersect.

Rewrite 4x - 6y = 12 as y = (2/3)x - 2, giving slope 2/3. For 2x + ky = 8, solving for y gives slope = -2/k. Set equal: -2/k = 2/3, so 2k = -6 and k = -3. Choice A (3) gives slope -2/3, which is the negative of 2/3, not equal to it.

Slope = (-4 - 4) / (3 - (-1)) = -8 / 4 = -2. Using point-slope form with (-1, 4): y - 4 = -2(x + 1), which simplifies to y = -2x + 2. Set y = 0: 2x = 2, so x = 1. The x-intercept is (1, 0). Choice C (2, 0) comes from forgetting to divide by 2 when solving.

Slope = (4 - 0) / (0 - (-6)) = 4/6 = 2/3. The y-intercept is 4, giving y = (2/3)x + 4. Choice B incorrectly negates the slope. Moving from (-6, 0) to (0, 4) goes right 6 and up 4, confirming a positive slope of 2/3.

Set up the slope equation: (k - 3) / (1 - k) = -2. Cross multiply: k - 3 = -2(1 - k) = -2 + 2k. Subtract 2k from both sides: -k - 3 = -2, so -k = 1 and k = -1. Verify: points (-1, 3) and (1, -1) give slope (-1 - 3)/(1 - (-1)) = -4/2 = -2. Choice B (1) results from a sign error when distributing -2.

Rewrite 4x - 8y = 16 as y = (1/2)x - 2, so slope = 1/2. The perpendicular slope is -2 (negative reciprocal of 1/2). Since the line passes through the origin, b = 0, giving y = -2x. Choice B has the correct slope but passes through (0, 2) instead of the origin. Choice A is parallel to the given line, not perpendicular.

Rewrite 6x - 4y = 8 as y = (3/2)x - 2, giving slope 3/2. The parallel line with y-intercept 3 is y = (3/2)x + 3. Set y = 0: (3/2)x = -3, so x = -2. The line crosses the x-axis at (-2, 0). Choice A (2, 0) comes from dropping the negative sign when solving for x.

Slope measures how much y changes for each unit increase in x — the rate of change. The x-intercept is where the line crosses the x-axis, the y-intercept is where it crosses the y-axis, and the length between two points requires the distance formula, not slope.

In slope-intercept form y = mx + b, m is the slope (rate of change) and b is the y-intercept. Confusing m and b is one of the most common errors when reading slope-intercept form.

A horizontal line has no vertical change (rise = 0) for any horizontal run, so slope = rise/run = 0/run = 0. A vertical line has undefined slope because the run equals zero, making division impossible. Confusing horizontal and vertical slopes is a classic error.

In slope-intercept form y = mx + b, the slope is the coefficient of x. Here m = -4 and b = 7. Choosing 7 or -7 confuses the y-intercept with the slope.

A line has a positive slope when it rises from left to right. In y = 4x - 1, the coefficient of x is 4, which is positive. All other choices have negative coefficients of x (-3, -1, -0.5), meaning those lines fall from left to right.

In slope-intercept form y = mx + b, b is the y-intercept. Here b = -10, so the line crosses the y-axis at (0, -10). Choosing 6 confuses the slope with the y-intercept.

A vertical line has the same x-value for all its points, so the run (change in x) is always 0. Since slope = rise/run and division by zero is undefined, vertical lines have undefined slope. Horizontal lines have slope 0, which is defined — not undefined.

Slope = (y2 - y1)/(x2 - x1) = (13 - 5)/(6 - 2) = 8/4 = 2. A common error is inverting the formula to get (6 - 2)/(13 - 5) = 1/2. The value 8 is only the numerator (rise) before dividing by the run.

Using slope-intercept form y = mx + b with m = 3 and b = -5 gives y = 3x - 5. Choice A swaps the slope and y-intercept values. Choice D uses +5 instead of -5 for the y-intercept.

Solve for y: subtract 2x to get -6y = -2x + 12, then divide by -6 to get y = (1/3)x - 2. The slope is 1/3. A common error is reading the coefficient of x before isolating y, which incorrectly gives a slope of 2.

Since (0, 8) is on the line, the y-intercept is 8. Slope = (8 - 2)/(0 - (-3)) = 6/3 = 2. The equation is y = 2x + 8. Choice D incorrectly computes the slope as 1/2 by swapping rise and run.

Steepness is determined by the absolute value of the slope. The slopes are 1/2 = 0.5, 2/3 ≈ 0.67, 3/4 = 0.75, and 1. Since |1| is the largest, y = x - 5 is the steepest of the four lines.

Perpendicular lines have slopes that are negative reciprocals of each other. The slope of the given line is 1/3, so the perpendicular slope is -1/(1/3) = -3. Choice C (3) is the reciprocal but is missing the required negative sign.

Two distinct lines are parallel when they have equal slopes and different y-intercepts. Both lines have slope 4, and their y-intercepts (-2 and 7) differ, so they are parallel. For perpendicular lines the slopes would need to multiply to -1, which 4 times 4 does not satisfy.

Rate of change = (225 - 90)/(5 - 2) = 135/3 = 45 miles per hour. This is the slope of the distance-time line. Choice B (135) is the total additional miles traveled across those 3 hours, not the hourly rate.

The x-intercept occurs when y = 0: set 0 = -2x + 8, so 2x = 8 and x = 4. The x-intercept is (4, 0). Choice A (-4) results from incorrectly treating -2x = 8 without accounting for the positive 8 on the other side.

Slope from A to B = (5 - 1)/(2 - 0) = 2. Slope from B to C = (9 - 5)/(4 - 2) = 2. Equal slopes confirm all three points lie on the same line (y = 2x + 1), making them collinear. Collinear points do not require equal x- or y-values.

Slope = (3 - (-1))/(8 - 4) = 4/4 = 1. Using point-slope form with (4, -1): y + 1 = 1(x - 4), so y = x - 5. The y-intercept is -5. Choice D (-1) is the y-value of the first given point, not where the line crosses the y-axis.

A parallel line has the same slope, so m = -3. Using point-slope form: y - 4 = -3(x - 2), giving y = -3x + 6 + 4 = -3x + 10. Choice C incorrectly treats the y-value of the given point (4) as the y-intercept without adjusting for the x-coordinate.

Rewrite 3x + y = 9 as y = -3x + 9, so the slope is -3. The perpendicular slope is the negative reciprocal: 1/3. Using point-slope with (3, -1): y + 1 = (1/3)(x - 3), so y = (1/3)x - 1 - 1 = (1/3)x - 2. Choice B uses the original slope (-3) rather than the perpendicular slope.

Line 2: -4y = -2x + 8, so y = (1/2)x - 2 and slope = 1/2. For perpendicular lines, slopes must multiply to -1. Line 1 has slope -3/k, so (-3/k)(1/2) = -1, giving -3 = -2k and k = 3/2. Choosing k = 6 gives Line 1 a slope of -1/2, and (-1/2)(1/2) = -1/4, which is not -1.

Slope = (-2 - 4)/(4 - (-2)) = -6/6 = -1. Using point-slope with (-2, 4): y - 4 = -1(x + 2), so y = -x + 2. Setting y = 0: x = 2. Choice B (-2) is the x-coordinate of the first given point, not the x-intercept of the line.

Find the x-intercept: 0 = (3/4)x - 6, so x = 8. The x-intercept is (8, 0). The y-intercept is (0, -6). The base along the x-axis is 8 units and the height along the y-axis is 6 units. Area = (1/2)(8)(6) = 24. Choice A (12) results from forgetting to multiply by 1/2 in the triangle area formula.

Find the slope using (4, 6) and (6, 10): slope = (10 - 6)/(6 - 4) = 4/2 = 2. For collinearity, the slope from (2, k) to (4, 6) must also equal 2: (6 - k)/(4 - 2) = 2, so 6 - k = 4 and k = 2. Choosing k = 4 gives slope (6 - 4)/2 = 1, which does not match the required slope of 2.

The perpendicular slope to 3 is -1/3. Using point-slope form for line q through (6, 2): y - 2 = -1/3(x - 6), so y = -1/3x + 2 + 2 = -1/3x + 4. The y-intercept is 4. Choice C (2) is the y-coordinate of the given point on line q, not the y-intercept. Choice B (-4) results from a sign error when distributing -1/3 through (x - 6).

In slope-intercept form y = mx + b, m is the slope. For y = 5x - 3, the coefficient of x is 5, so the slope is 5. The value -3 is the y-intercept, not the slope — a very common mix-up.

In y = mx + b, b is the y-intercept — where the line crosses the y-axis. Here b = 7, so the y-intercept is 7, meaning the line passes through (0, 7). The value -2 is the slope m, not the y-intercept.

In y = mx + b, b is the y-intercept — the y-value where the line crosses the y-axis, written as the point (0, b). The slope is m, also called the rate of change. The x-intercept is found by setting y = 0 and solving for x, not by reading b directly.

Slope-intercept form is y = mx + b, requiring y to be isolated on the left side. Only y = 4x - 1 matches this pattern directly. The other choices are in standard form or have x isolated rather than y.

A horizontal line has equation y = c, meaning y never changes as x increases. Slope = rise/run = 0/run = 0. An undefined slope belongs to vertical lines, where the run equals 0, causing division by zero — not to horizontal lines.

A vertical line has equation x = c. When calculating slope, the run (change in x) equals 0. Since division by zero is undefined, a vertical line has no defined slope. A slope of 0 belongs to horizontal lines, where the rise equals 0.

The equation y = 3x can be rewritten as y = 3x + 0. In slope-intercept form y = mx + b, the y-intercept is b = 0, meaning the line passes through the origin (0, 0). The value 3 is the slope, not the y-intercept.

Substitute each x-value into y = 2x + 1: (1, 4): 2(1)+1 = 3, not 4. (2, 5): 2(2)+1 = 5, which matches. (3, 8): 2(3)+1 = 7, not 8. (0, 2): 2(0)+1 = 1, not 2. Only (2, 5) satisfies the equation.

Substitute into y = mx + b with m = -2 and b = 5 to get y = -2x + 5. Choice A swaps the slope and y-intercept. Choice C uses a positive slope of 2. Choice D uses -5 as the slope rather than -2.

Slope = (y2 - y1) / (x2 - x1) = (11 - 3) / (5 - 1) = 8 / 4 = 2. A common error is dividing the change in x by the change in y, giving 4/8 = 0.5. Always place the change in y in the numerator.

We need m < 0 and b > 0. For y = -x + 4: m = -1 (negative) and b = 4 (positive). Choice A has positive slope 3. Choice C has positive slope 2. Choice D has a negative slope but also a negative y-intercept of -2.

The x-intercept occurs where y = 0. Set 0 = 3x - 9, then add 9 to both sides: 3x = 9, so x = 3. The x-intercept is (3, 0). The point (0, -9) is the y-intercept — a frequent error is confusing which variable equals zero for each intercept.

Use point-slope form: y - 5 = 3(x - 2). Distribute: y - 5 = 3x - 6. Add 5 to both sides: y = 3x - 1. Choice A comes from incorrectly adding 6 to 5 to get b = 11. Choice C mistakenly uses the y-coordinate of the given point directly as b.

Rewrite in slope-intercept form by isolating y: subtract 2x from both sides to get y = -2x + 8. The slope is the coefficient of x, which is -2. Choice A uses the coefficient of x from the original equation without recognizing the subtraction. Choice D divides 8 by 2.

Perpendicular lines have slopes that are negative reciprocals, meaning m1 times m2 = -1. For choice B: 3 times (-1/3) = -1, so they are perpendicular. Choice A: both slopes equal 2 — those lines are parallel. Choice C: both slopes equal 1 — parallel. Choice D: 5 times (-5) = -25, not -1.

A line passes through the origin (0, 0) when its y-intercept b = 0. For y = -5x, we can write y = -5x + 0, so b = 0 and (0, 0) satisfies the equation. The other choices have y-intercepts of 1, 3, and 2 respectively, so none of them pass through the origin.

Using the two intercept points (3, 0) and (0, -6): slope = (0 - (-6)) / (3 - 0) = 6 / 3 = 2. Choice A (-2) is a sign error that results from computing the y-intercept value divided by the x-intercept value: -6/3 = -2, which ignores the direction of travel along the line.

First rewrite 4x - 2y = 6 as y = 2x - 3; the slope is 2. The perpendicular slope is -1/2. Apply point-slope form through (2, -1): y + 1 = -1/2(x - 2), so y + 1 = -1/2 x + 1, giving y = -1/2 x. Choice A incorrectly gets b = 2 by not distributing -1/2 correctly.

The intercepts give two points: (4, 0) and (0, -3). Slope = (0 - (-3)) / (4 - 0) = 3/4. The y-intercept is already known as -3, so the equation is y = (3/4)x - 3. Choice B negates the slope incorrectly. Choices C and D flip the numerator and denominator of the slope fraction.

Slope of m: (9-5)/(3-1) = 4/2 = 2. Equation: y = 2x + 3. Slope of n: (4-0)/(1-(-1)) = 4/2 = 2. Equation: y = 2x + 2. Both have slope 2 but different y-intercepts (3 and 2), so they are parallel. They are not the same line, since the y-intercepts differ — ruling out choice A.

Substitute the point and slope into y = mx + b: 7 = 3(2) + b, so 7 = 6 + b, giving b = 1. Choice C (13) comes from adding 6 to 7 instead of subtracting: 7 + 6 = 13. Choice A (4) comes from subtracting the slope itself: 7 - 3 = 4, which skips substituting x correctly.

Rewrite 2x - 5y = 10 as y = (2/5)x - 2, giving slope 2/5. The parallel line has the same slope. Using point-slope through (5, 1): y - 1 = (2/5)(x - 5), so y - 1 = (2/5)x - 2, giving y = (2/5)x - 1. The y-intercept is -1. Choice B (2) is the y-intercept of the original line, not the new parallel line.

Since lines A and B are parallel, they share the same slope: -(1/4). A line perpendicular to B has the negative reciprocal slope: -1 / (-1/4) = 4. Choice B (1/4) flips the fraction but forgets to negate it. Choice A (-4) negates but forgets to flip. Both steps are required.

Slope of AB: (8-2)/(4-1) = 6/3 = 2. Perpendicular slope = -1/2. Apply point-slope form through B(4, 8): y - 8 = -1/2(x - 4), so y - 8 = -1/2 x + 2, giving y = -1/2 x + 10. Choice C (y = -1/2 x + 8) uses the correct slope but incorrectly treats the y-coordinate of B as the y-intercept.

Substitute (2, 6) into y = mx - 4: 6 = m(2) - 4. Add 4 to both sides: 10 = 2m. Divide by 2: m = 5. Choice B (m = 3) comes from ignoring the -4 and dividing y by x: 6/2 = 3. Choice D (m = 7) comes from adding 4 to 6 to get 10 but then making an arithmetic error.

In slope-intercept form y = mx + b, m is the slope and b is the y-intercept. Here m = 4, so the slope is 4. Choice A (-7 is actually the y-intercept, not the slope. Choice D (1/4) is the reciprocal of the slope, a common error when students confuse slope with its inverse.

In slope-intercept form y = mx + b, b is the y-intercept. Here b = 5, so the y-intercept is 5. Choice A (-2) is the slope, not the y-intercept. The y-intercept is the constant term added to the slope term.

A horizontal line has the equation y = c for some constant c. The slope formula gives (c - c)/(x2 - x1) = 0/change-in-x = 0. Choice D (undefined) describes vertical lines, not horizontal ones. A horizontal line has zero rise over any run.

A vertical line has the equation x = c. Any two points on it share the same x-value, so the slope formula gives (y2 - y1)/(c - c) = change-in-y/0, which is division by zero and therefore undefined. Choice A (0) describes horizontal lines. Vertical lines have undefined slope because the run is always zero.

Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Choice B matches this exactly with m = 5 and b = -1. Choice A is standard form, Choice C is point-slope form, and Choice D is intercept form.

Substituting m = -3 and b = 2 into y = mx + b gives y = -3x + 2. Choice A swaps the slope and y-intercept values. Choice B uses the correct slope but the wrong sign for the y-intercept. Choice D uses the correct y-intercept but the wrong sign for the slope.

In slope-intercept form y = mx + b, the sign of m determines the direction of the line. Only Choice C has a negative coefficient for x (m = -5), meaning the line falls from left to right. All other choices have positive slopes, so those lines rise from left to right.

Slope = (y2 - y1)/(x2 - x1) = (11 - 3)/(4 - 0) = 8/4 = 2. Choice C (3) is a common error from subtracting 11 - 8 = 3 or misreading the coordinates. Choice D (4) is just the x-value of the second point, not the slope.

Substitute into y = mx + b: 8 = 5(1) + b, so 8 = 5 + b, giving b = 3. Choice D (13) results from adding 5 and 8 instead of solving for b. Choice C (5) is the slope, not the y-intercept.

Parallel lines have equal slopes. The given line has slope 3, so any parallel line must also have slope 3. Only Choice C has slope 3 (with a different y-intercept, confirming the lines are distinct). Choice B uses the reciprocal slope 1/3, and Choice A uses the negative slope -3, which is the slope of a perpendicular line.

To find the x-intercept, set y = 0: 0 = -2x + 8, so 2x = 8, giving x = 4. The x-intercept is 4. Choice C (8) is the y-intercept, found by setting x = 0. Choice A (-4) results from a sign error when solving: -2x = 8 would give x = -4, but the equation requires subtracting 8 from both sides first.

Rewrite in slope-intercept form: subtract 3x from both sides to get -y = -3x + 6, then multiply by -1 to get y = 3x - 6. The slope is 3. Choice D (6) is the constant on the right side of the original equation, not the slope. Choice B (-3) results from forgetting to multiply both sides by -1 when solving for y.

Perpendicular lines have slopes that are negative reciprocals of each other. The given slope is 1/2, so the perpendicular slope is -1/(1/2) = -2. Only Choice C has slope -2. Choice A has slope 2 (the reciprocal but not the negative reciprocal). Choice B has the same slope as the original line, making it parallel, not perpendicular.

Slope = (y2 - y1)/(x2 - x1) = (0 - 4)/(2 - (-2)) = -4/4 = -1. The slope is -1. Choice A (2) results from dividing the change in x by the change in y. Choice B (-2) results from computing -4/2 instead of -4/4, forgetting to account for the negative x-value in the denominator.

Solve for y: 2y = -4x + 12, so y = -2x + 6. The y-intercept is 6. Choice A (3) results from dividing 12 by 4 instead of by 2. Choice D (12) is the constant in the original equation before dividing. Alternatively, set x = 0: 2y = 12, so y = 6.

A line through the origin has b = 0 in y = mx + b, giving y = mx. With slope m = -4, the equation is y = -4x. Choice A (y = -4) is a horizontal line with slope 0, not -4. Choice B (x = -4) is a vertical line that does not pass through the origin. Choice D has the correct form but the wrong sign for the slope.

Substitute into y = mx + b: 4 = 2(3) + b, so 4 = 6 + b, giving b = 4 - 6 = -2. The y-intercept is -2. Choice C (10) results from adding 4 + 6 instead of finding b = 4 - 6. Choice B (2) is the slope of the line, not the y-intercept.

The two intercepts give points (-2, 0) and (0, 4). Slope = (4 - 0)/(0 - (-2)) = 4/2 = 2. With y-intercept 4, the equation is y = 2x + 4. Choice B uses the x-intercept value as the slope. Choice C uses the correct slope but a wrong sign for the y-intercept. Choice D uses the reciprocal slope.

The perpendicular slope is the negative reciprocal of 3/4, which is -4/3. Using point-slope: -1 = (-4/3)(3) + b, so -1 = -4 + b, giving b = 3. The y-intercept is 3. Choice A (-4) is the product (-4/3)(3) without adding it to -1. Choice B (-1) is the y-coordinate of the given point, not the y-intercept.

Slope of p: (13 - 5)/(6 - 2) = 8/4 = 2. Slope of q: (5 - (-3))/(4 - 0) = 8/4 = 2. The slopes are equal, so the lines are parallel. They are not the same line because p has y-intercept 1 (from 5 = 2(2) + b) while q has y-intercept -3. Since the slopes are equal (not negative reciprocals), Choice C is incorrect.

Rewrite kx - 3y = 9 as y = (k/3)x - 3, giving slope k/3. For perpendicularity, the product of slopes must equal -1: (k/3)(3/4) = -1, so k/4 = -1, giving k = -4. Choice B (4) gives a slope of 4/3, whose product with 3/4 is 1, not -1, so lines with these slopes are parallel to a situation where you'd need k = -4. Choice C (-9) would make the slope -3, not -4/3.

Slope of the first line: (11 - 3)/(5 - 1) = 8/4 = 2. The parallel line has the same slope 2. Using point (2, 1): 1 = 2(2) + b, so 1 = 4 + b, giving b = -3. Choice C (1) is the y-coordinate of the given point, not the y-intercept. Choice A (-5) results from computing 1 - 2(3) instead of 1 - 2(2).

Rewrite Line A: 2y = 6x - 10, so y = 3x - 5. Slope is 3 and y-intercept is -5. Line B has the same slope (3) and y-intercept -5 + 3 = -2. So Line B: y = 3x - 2. Choice B adds 3 to the slope instead of the y-intercept. Choice D subtracts 3 instead of adding it: -5 - 3 = -8.

Find the slope from A to B: (6 - 2)/(3 - 1) = 4/2 = 2. For C to be collinear, the slope from A to C must also equal 2: (k - 2)/(5 - 1) = 2, so (k - 2)/4 = 2, giving k - 2 = 8, so k = 10. Choice A (8) results from forgetting to add 2: solving k = 8 without the +2. Choice C (12) results from using slope from B to C incorrectly.

Let the x-intercept equal a, so the y-intercept is 3a. Using intercept form x/a + y/(3a) = 1, substitute (2, 6): 2/a + 6/(3a) = 1, so 2/a + 2/a = 1, giving 4/a = 1, so a = 4. The x-intercept is 4 and y-intercept is 12. Slope = (12 - 0)/(0 - 4) = -3. Choice B (3) uses the magnitude but forgets the negative sign since the line falls from y-intercept to x-intercept moving right.

In slope-intercept form y = mx + b, m is the slope and b is the y-intercept. In y = -3x + 7, the coefficient of x is -3, so the slope is -3. The value 7 is the y-intercept, not the slope — a common mix-up when reading the equation.

The slope is the coefficient of x in slope-intercept form. Only y = -3x + 2 has a negative coefficient (-3), meaning the line falls from left to right. All other choices have positive slopes (4, 1, and 2/3 respectively), so those lines rise from left to right.

A vertical line has the form x = c. Slope is defined as rise divided by run, but a vertical line has a run of 0 for any two points on it, making division by zero impossible. Therefore the slope is undefined. This is a common confusion with horizontal lines, which have slope 0 (zero rise, nonzero run).

Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Substituting m = -3 and b = 4 gives y = -3x + 4. Choice A swaps the slope and y-intercept. Choice B uses the correct slope but incorrectly makes the y-intercept negative.

Solve for y to get slope-intercept form: subtract 2x from both sides to get 4y = -2x + 8, then divide by 4 to get y = -(1/2)x + 2. The slope is the coefficient of x, which is -1/2. A common error is reading the coefficient of x directly from the standard form (2) and forgetting to divide by 4 and account for the negative sign.

Parallel lines have equal slopes and different y-intercepts. Both y = 2x - 5 and y = 2x + 3 have slope 2 but different y-intercepts (-5 and 3), confirming they are parallel and never intersect. Choice A has slopes of 3 and -3 — opposite in sign, so those lines are not parallel.

Since the line passes through (0, 6), the y-intercept is 6 and the equation is y = -(3/2)x + 6. Set y = 0 to find the x-intercept: 0 = -(3/2)x + 6, so (3/2)x = 6, giving x = 4. Choice C (x = 9) is a frequent error resulting from dividing 6 by 2/3 instead of the correct slope magnitude 3/2.

First rewrite line m in slope-intercept form: 2x - 5y = 10 becomes y = (2/5)x - 2, so its slope is 2/5. Since line n is parallel, it has the same slope 2/5. Applying point-slope form with (5, 1): y - 1 = (2/5)(x - 5), which simplifies to y = (2/5)x - 2 + 1 = (2/5)x - 1. The y-intercept is -1. A common mistake is using line m's y-intercept (-2) without adjusting for the new point.

First find the slope of the line through (-3, 1) and (1, -7): m = (-7 - 1) / (1 - (-3)) = -8/4 = -2. The slope of any perpendicular line is the negative reciprocal: m_perp = -1/(-2) = 1/2. Choice D (-1/2) is a common error — it takes the reciprocal of -2 without flipping the sign. Both steps (flip and negate) are required.

Using the two intercept points (a, 0) and (0, b), slope = (b - 0) / (0 - a) = b / (-a). Substituting b = -2a gives (-2a) / (-a) = 2. Because a cancels completely, the slope is always 2 no matter what nonzero value a takes. Choice C is incorrect because the variable a cancels out, making the slope a fixed constant — this is the key insight that makes this a synthesis-level problem.