➕ Algebra 1 · Linear Functions

Parallel and Perpendicular Lines practice questions

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Answer key with explanations

1. Which line is parallel to y = 4x − 9?

   Answer: y = 4x + 1. Parallel lines have the same slope. The given slope is 4, and only y = 4x + 1 has slope 4, with a different y-intercept (1 instead of −9). The line y = −(1/4)x − 9 is perpendicular, not parallel, because 4 × (−1/4) = −1. The lines y = −4x − 9 and y = (1/4)x − 9 are neither.

2. What is the slope of a line perpendicular to y = (5/3)x − 2?

   Answer: −3/5. Flip 5/3 to get 3/5, then change the sign: −3/5. Check: (5/3)(−3/5) = −15/15 = −1. The choice 3/5 flips but forgets the sign change (the product is 1). The choice −5/3 changes the sign but forgets the flip (the product is −25/9). The choice 5/3 is the same slope, which gives a parallel line.

3. Which equation is the line through (2, 9) that is parallel to y = 5x − 3?

   Answer: y = 5x − 1. Parallel means the same slope, 5, so y = 5x + b. Substitute (2, 9): 9 = 5(2) + b, so 9 = 10 + b and b = −1. The line is y = 5x − 1. Check: 5(2) − 1 = 9. The choice y = 5x + 9 uses the y-value 9 as b, but the point has x = 2, not 0. The choice y = 5x + 1 is a sign slip. The choice y = −5x + 19 passes through (2, 9) but has the wrong slope.

4. Which equation is the line through (6, 2) that is perpendicular to y = 3x + 4?

   Answer: y = −(1/3)x + 4. The given slope is 3, which is 3/1. Flip it to 1/3 and change the sign: −1/3. Substitute (6, 2) into y = −(1/3)x + b: 2 = −(1/3)(6) + b = −2 + b, so b = 4. The line is y = −(1/3)x + 4. The line y = (1/3)x passes through (6, 2) but only flipped the slope. The line y = −3x + 20 passes through (6, 2) but only changed the sign. The line y = −(1/3)x + 2 has the right slope but uses the y-value 2 as b: at x = 6 it gives 0.

5. Which describes the lines y = 2x − 5 and y = x/2 + 1?

   Answer: Neither parallel nor perpendicular. The slopes are 2 and 1/2. They are not equal, so the lines are not parallel. The product is 2 × (1/2) = 1, not −1, so they are not perpendicular. They are reciprocals, but a perpendicular partner also needs the sign change: the partner of 2 is −1/2.

6. Which describes the lines 6x + 2y = 5 and y = −3x + 1?

   Answer: Parallel. Solve the first for y: 2y = −6x + 5. Divide every term by 2: y = −3x + 5/2. Its slope is −3 and its y-intercept is 5/2. The second line has slope −3 and y-intercept 1. Same slope, different y-intercepts: parallel. (Reading the slope of the first line as 6, the number in front of x, is the trap.)

7. Which statement is true about the lines x = 3 and x = −8?

   Answer: They are parallel, because both are vertical lines.. Every point on x = 3 has x-coordinate 3, and every point on x = −8 has x-coordinate −8, so these are two different vertical lines. Two different vertical lines never meet, so they are parallel. Their slopes are both undefined, so there are no slope numbers to compare, but the lines can still be compared by their picture.

8. Which equation is the line through (−4, 3) that is perpendicular to 2x − 5y = 10?

   Answer: y = −(5/2)x − 7. Solve for y to find the slope: −5y = −2x + 10, so y = (2/5)x − 2. The slope is 2/5. Flip it to get 5/2 and change the sign: −5/2. Substitute (−4, 3) into y = −(5/2)x + b: 3 = −(5/2)(−4) + b = 10 + b, so b = −7. The line is y = −(5/2)x − 7. The choice y = (5/2)x + 13 passes through the point but has slope 5/2 (the sign change is missing). The choice y = −(5/2)x + 3 uses the y-value 3 as b. The choice y = −(5/2)x + 13 is a sign slip in solving for b.

9. Which is the equation of the line through (5, −2) that is perpendicular to y = 8?

   Answer: x = 5. y = 8 is horizontal, so a perpendicular line is vertical. A vertical line has an equation x = a. It must pass through (5, −2), so every point on it has x-coordinate 5: x = 5. The line y = −2 is horizontal, so it is parallel to y = 8. The line x = −2 swaps the coordinates. The line y = 8 is the given line itself. (The 8 in y = 8 is the y-intercept, not a slope to flip.)

10. A hiking trail follows the line y = (1/4)x + 1 on a map where both axes use the same scale. A new trail must be perpendicular to it and start at the lodge at (3, 10). Which equation describes the new trail?

    Answer: y = −4x + 22. The given slope is 1/4. Flip it to get 4 and change the sign: −4. Substitute (3, 10) into y = −4x + b: 10 = −4(3) + b = −12 + b, so b = 22. The trail is y = −4x + 22. Check: (1/4)(−4) = −1. The choice y = 4x − 2 passes through the lodge but flipped without the sign change. The choice y = −4x + 10 uses the y-value 10 as b. The choice y = −4x − 22 is a sign slip in solving for b. The same-scale detail matters: only then does a right angle on the map match a right angle on the ground.