HighSchoolQ

➕ Algebra 1 · Linear Functions

Parallel and Perpendicular Lines

Tell whether two lines are parallel, perpendicular or neither from their slopes, and write the equation of the line through a point that is parallel or perpendicular.

Grades 9, 10⏱ 16 min✔ AI-checkedCCSS G-GPE.B.5
About standards and checking

Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Three blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-07.

  • G-GPE.B.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

By the end you can…

  • State the slope rules: parallel lines have the same slope and different y-intercepts, and perpendicular lines have slopes that are negative reciprocals (their product is −1).
  • Handle horizontal and vertical lines, which need their own rule because 0 has no reciprocal and a vertical line has an undefined slope.
  • Decide whether two lines are parallel, perpendicular or neither, including equations that must be rearranged into slope-intercept form first.
  • Write the equation of the line through a given point that is parallel or perpendicular to a given line.

Lesson (school style)

Two different lines in a plane either never meet, which makes them parallel, or they cross at exactly one point. If they cross at a right angle (90 degrees), they are perpendicular. You do not need a ruler or a protractor to tell which: the slopes give it away. This lesson shows the two slope rules (the same slope for parallel, negative reciprocals for perpendicular), the special case of horizontal and vertical lines, and how to use the rules to write the equation of a new line through a given point.

Two lines, three possibilities

Take two different lines in a coordinate plane. Exactly one of these is true:

  • They never meet: the lines are parallel.
  • They cross at exactly one point at a right angle: the lines are perpendicular.
  • They cross at exactly one point at some other angle: the lines are neither parallel nor perpendicular.

In this lesson, parallel lines are two different lines. A line is not called parallel to itself. If two equations turn out to describe the same line, we say “the same line.”

Parallel lines: same slope, different y-intercept

Non-vertical parallel lines tilt the same way by the same amount, so they have the same slope m. They also cross the y-axis at different places, so they have different y-intercepts. For example, y = 2x + 1 and y = 2x − 4 are parallel.

If both the slope and the y-intercept matched, the two equations would describe the same line, not two parallel lines.

Why do they never meet? Take two non-vertical lines y = mx + b1 and y = mx + b2 with b1 ≠ b2. Suppose they met at a point. At that point both equations give the same y, so mx + b1 = mx + b2. Subtract mx from both sides: b1 = b2. But the y-intercepts are different, so that is impossible. The lines never meet.

And if two non-vertical lines have different slopes, m1 ≠ m2, they always meet at exactly one point. Set m1x + b1 = m2x + b2 and gather the x-terms: (m1 − m2)x = b2 − b1. The number m1 − m2 is not 0, so you can divide by it to get exactly one value of x.

Perpendicular lines: negative reciprocals

Two lines that are neither horizontal nor vertical are perpendicular exactly when their slopes are negative reciprocals of each other. That means the product of the slopes is −1.

To find a perpendicular slope, do two things:

  1. Flip the fraction (the reciprocal). Flipping keeps the sign.
  2. Change the sign.

For example, 2/3 becomes 3/2 after the flip, and −3/2 after the sign change. Check the product: (2/3)(−3/2) = −1. A whole number is a fraction over 1: write −5 as −5/1, flip it to 1/(−5), which is −1/5, then change the sign to get 1/5.

You need both steps. Changing only the sign (3 and −3) or only flipping (3 and 1/3) does not work. Since the product must be negative, one slope of a perpendicular pair is positive and the other is negative.

Why negative reciprocals? A quick reason

This is a short argument, not a full proof. Take the line through the origin with slope 2/5. The point (5, 2) is on it (run 5, rise 2). Turn the whole picture a quarter turn (90 degrees) counterclockwise about the origin. A quarter turn sends any point (x, y) to (−y, x), so (5, 2) lands on (−2, 5).

The turned line is perpendicular to the original line. It passes through the origin and (−2, 5), so its slope is 5 ÷ (−2) = −5/2. The rise and the run swapped places, and one of them changed sign. That is exactly “flip, then change the sign.” The same thing happens for any slope a/b with a and b not zero: the turned line has slope −b/a, and (a/b)(−b/a) = −1.

Horizontal and vertical lines

The flip-and-change-sign rule cannot handle every line. A horizontal line y = b has slope 0, and 0 has no reciprocal because 1 ÷ 0 is undefined. A vertical line x = a has an undefined slope, so there is no number to flip.

So these lines have their own rules:

  • A horizontal line and a vertical line are perpendicular. For example, y = 4 and x = −3 cross at a right angle at (−3, 4).
  • Only horizontal lines are perpendicular to a vertical line. A slanted line like y = 2x + 1 is not.
  • Two different vertical lines, like x = 2 and x = 7, are parallel. Two different horizontal lines, like y = 1 and y = 5, are parallel too (same slope 0, different y-intercepts).

Testing two lines

To decide whether two lines are parallel, perpendicular or neither:

  1. Look for vertical lines first (equations like x = a). Two different vertical lines are parallel. A vertical line and a horizontal line are perpendicular. A vertical line and a slanted line are neither.
  2. Write both equations in the form y = mx + b. If an equation is not solved for y, solve for y first.
  3. Compare the slopes and y-intercepts, as listed next.
  • Same slope, different y-intercepts: parallel.
  • Same slope, same y-intercept: the same line.
  • Product of the slopes is −1: perpendicular.
  • Anything else: neither.

Be careful with “anything else.” Slopes like 4 and −4 are opposites but not negative reciprocals (the product is −16). Slopes like 2 and 1/2 are reciprocals but not negative reciprocals (the product is 1).

Rearranging first

An equation like 2x + 3y = 9 is not in slope-intercept form, and its slope is not 2. Solve for y: subtract 2x from both sides to get 3y = −2x + 9, then divide every term by 3 to get y = −(2/3)x + 3. Now you can read the slope, −2/3.

Do not read a slope from the number in front of x until y is alone with a coefficient of 1. (For lines written as Ax + By = C with B ≠ 0, this always works out to slope −A/B.)

Writing a parallel or perpendicular line through a point

You are given a line and a point (x1, y1), and you need the new line through that point.

  1. Slope of the given line (solve for y if you need to).
  2. Look at what you need. Parallel: use the same slope. Perpendicular: flip the slope, then change the sign.
  3. Insert the point into y = mx + b with the new slope.
  4. Pull out b by solving, then write the equation and check it.

The point’s y-value is not b unless the point’s x is 0. If the given line is horizontal or vertical, skip the steps: a perpendicular to a horizontal line through (x1, y1) is the vertical line x = x1, and a perpendicular to a vertical line is the horizontal line y = y1.

Parallel and perpendicular in real life

Parallel lines have the same rate of change but different starting values. Two plans with the same price per hour and different sign-up fees give parallel cost lines: the gap between them never changes, and the plans never cost the same.

A caution about perpendicular: a right angle only looks like a right angle if both axes use the same units and the same scale, like on a map. On a graph of cost against hours, how square a corner looks depends on how you stretched the axes. The slopes tell the truth either way.

Key words

Parallel lines
Two different lines in a plane that never meet. Non-vertical parallel lines have the same slope and different y-intercepts.
Perpendicular lines
Two lines that cross at a right angle (90 degrees). Their slopes are negative reciprocals, unless one line is horizontal and the other vertical.
Reciprocal
The number you get by flipping a fraction. The reciprocal of 2/3 is 3/2, and the reciprocal of 5 is 1/5. The sign stays the same, and 0 has no reciprocal.
Negative reciprocal
The reciprocal with its sign changed. The negative reciprocal of 2/3 is −3/2. A number times its negative reciprocal is −1.
Intersect
To cross. Two lines intersect at the point where they cross.
Y-intercept
The number b in y = mx + b: the y-value where the line crosses the y-axis, at the point (0, b).

Worked examples

Example 1. Decide whether each pair of lines is parallel, perpendicular or neither. (a) y = 3x − 4 and y = 3x + 6. (b) y = (2/5)x + 1 and y = −(5/2)x − 3. (c) y = 4x + 2 and y = −4x + 2.

  1. (a) Both slopes are 3, and the y-intercepts are different (−4 and 6). Same slope, different y-intercepts: the lines are parallel.
  2. (b) The slopes are 2/5 and −5/2. Multiply: (2/5)(−5/2) = −10/10 = −1. The product is −1, so the lines are perpendicular.
  3. (c) The slopes are 4 and −4. They are not equal, so the lines are not parallel. The product is 4 × (−4) = −16, not −1, so they are not perpendicular. These slopes are opposites but not flipped. The lines are neither. (They do cross, at (0, 2), but not at a right angle.)

Answer: Answer: (a) Parallel. (b) Perpendicular. (c) Neither.

Example 2. Find the slope of a line perpendicular to a line with slope (a) 7/3, (b) −6, (c) −1/4.

  1. (a) Flip 7/3 to get 3/7, then change the sign: −3/7. Check: (7/3)(−3/7) = −1.
  2. (b) Write −6 as −6/1. Flipping gives 1/(−6) = −1/6. Change the sign: 1/6. Check: (−6)(1/6) = −1.
  3. (c) Flip −1/4 to get −4/1 = −4 (flipping keeps the sign). Change the sign: 4. Check: (−1/4)(4) = −1.

Answer: Answer: (a) −3/7. (b) 1/6. (c) 4.

Example 3. (a) Are 2x + 3y = 9 and 3x − 2y = 4 parallel, perpendicular or neither? (b) Are y = 2x + 3 and 4x − 2y = −6 parallel?

  1. (a) Solve the first equation for y: 3y = −2x + 9. Divide every term by 3: y = −(2/3)x + 3. The slope is −2/3.
  2. Solve the second equation for y: −2y = −3x + 4. Divide every term by −2: y = (3/2)x − 2. The slope is 3/2.
  3. Multiply the slopes: (−2/3)(3/2) = −1. The lines are perpendicular.
  4. (b) Solve 4x − 2y = −6 for y: −2y = −4x − 6. Divide every term by −2: y = 2x + 3.
  5. That is exactly the first equation, y = 2x + 3. The slopes match and so do the y-intercepts, so these are the same line, not a pair of parallel lines.

Answer: Answer: (a) Perpendicular. (b) No: it is the same line.

Example 4. Decide whether each pair is parallel, perpendicular or neither. (a) x = 2 and x = 7. (b) y = 4 and x = −3. (c) x = 8 and y = 2x + 1.

  1. (a) Both lines are vertical, and they are different lines (one is at x = 2, the other at x = 7). 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 picture settles it.
  2. (b) y = 4 is horizontal (slope 0) and x = −3 is vertical (undefined slope). A horizontal line and a vertical line cross at a right angle, here at (−3, 4). They are perpendicular.
  3. (c) x = 8 is vertical. y = 2x + 1 has slope 2, so it is slanted, not horizontal. A slanted line crosses a vertical line at an angle that is not 90 degrees, so the lines are neither parallel nor perpendicular.

Answer: Answer: (a) Parallel. (b) Perpendicular. (c) Neither.

Example 5. Write the equation of the line through (4, 5) that is parallel to y = −2x + 7.

  1. Parallel lines have the same slope, so the new slope is −2: y = −2x + b.
  2. Substitute the point (4, 5): 5 = −2(4) + b, so 5 = −8 + b.
  3. Add 8 to both sides: b = 13.
  4. The equation is y = −2x + 13.
  5. Check. At x = 4: −2(4) + 13 = 5, so the line passes through (4, 5). The slope matches the given line, and the y-intercepts differ (13 and 7), so this is a different line, parallel to the first.

Answer: Answer: y = −2x + 13

Example 6. Write the equation of the line through (8, 3) that is perpendicular to y = (4/5)x − 6.

  1. The given slope is 4/5. Flip it to get 5/4, then change the sign: −5/4. So y = −(5/4)x + b.
  2. Substitute (8, 3): 3 = −(5/4)(8) + b. Since (5/4)(8) = 10, this is 3 = −10 + b.
  3. Add 10 to both sides: b = 13.
  4. The equation is y = −(5/4)x + 13.
  5. Check. The slopes multiply to (4/5)(−5/4) = −1. At x = 8: −(5/4)(8) + 13 = −10 + 13 = 3.

Answer: Answer: y = −(5/4)x + 13

Example 7. Write the equation of the line through (−3, 1) that is perpendicular to 3x − 4y = 12.

  1. The given equation is not solved for y. Subtract 3x from both sides: −4y = −3x + 12. Divide every term by −4: y = (3/4)x − 3. The slope is 3/4.
  2. Flip 3/4 to get 4/3, then change the sign: −4/3. So y = −(4/3)x + b.
  3. Substitute (−3, 1): 1 = −(4/3)(−3) + b. A negative times a negative is positive, so −(4/3)(−3) = 4 and 1 = 4 + b.
  4. Subtract 4 from both sides: b = −3. The equation is y = −(4/3)x − 3.
  5. Check. The slopes multiply to (3/4)(−4/3) = −1. At x = −3: −(4/3)(−3) − 3 = 4 − 3 = 1.

Answer: Answer: y = −(4/3)x − 3

Example 8. Write the equation of the line through (−2, 6) that is (a) perpendicular to y = 9, (b) perpendicular to x = 1.

  1. (a) y = 9 is horizontal, so a perpendicular line is vertical. Vertical lines have equations x = a. The line must pass through (−2, 6), so every point on it has x-coordinate −2: x = −2.
  2. (b) x = 1 is vertical, so a perpendicular line is horizontal. Horizontal lines have equations y = b. Every point on this line has y-coordinate 6: y = 6.
  3. Check. (−2, 6) is on both lines. The flip-and-change-sign rule was not needed here, and it could not be used: 0 has no reciprocal, and a vertical line’s slope is undefined.

Answer: Answer: (a) x = −2. (b) y = 6.

Example 9. Tutor A charges a 15-dollar sign-up fee plus 12 dollars per hour. Tutor B charges a 25-dollar sign-up fee plus 12 dollars per hour. Let h be the number of hours and C the total cost in dollars. Write an equation for each tutor. Are the lines parallel, and what does that tell you?

  1. Tutor A: C = 12h + 15. Tutor B: C = 12h + 25.
  2. Both slopes are 12 dollars per hour. The y-intercepts are 15 and 25, which are different. Same slope and different y-intercepts: the lines are parallel.
  3. Parallel lines never meet, so the two tutors never cost the same amount. Tutor B costs 10 dollars more at every number of hours: (12h + 25) − (12h + 15) = 10.
  4. Only h ≥ 0 makes sense in this story.

Answer: Answer: Parallel. Tutor B always costs 10 dollars more.

In plain English

Slope tells you how steep a line is. Two lines that are exactly as steep and tilt the same way will never meet. Think of the two rails of a straight railroad track. Those are parallel lines: the same slope, and they never meet.

Now picture a plus sign. One line goes left and right. The other goes up and down. They cross in a perfect square corner. Those lines are perpendicular. Tilt the whole plus sign and both lines tilt, but the corner stays square. Their slopes change in a special way: flip the fraction, then change the sign. A line with slope 2/3 gets a partner with slope −3/2.

To find out whether two lines are parallel, perpendicular or neither, first look for vertical lines like x = 2. Two different vertical lines are parallel. A vertical line and a horizontal line are perpendicular. A vertical line and a slanted line are neither. If neither line is vertical, put each equation in the form y = mx + b and compare the m numbers. The same m with a different b means parallel. The same m and the same b means the same line. Slopes that multiply to −1 mean perpendicular. Anything else means neither.

To build a new line through a point, keep the slope for parallel, or flip it and change the sign for perpendicular. Then plug the point into y = mx + b and solve for b.

See it

xy 246 246 run 2rise 1 run 1down 2 b = 1b = −2 Parallel: same slope y = (1/2)x + 1 y = (1/2)x − 2 Same slope 1/2, different b: they never meet. Perpendicular: product −1 y = (1/2)x + 1 y = −2x + 6 (1/2) × (−2) = −1 They cross at (2, 2) at a right angle. Both axes use the same scale, so a right angle looks like one.
Parallel lines share a slope (1/2). Perpendicular lines have slopes 1/2 and −2, which multiply to −1.
Write both lines as y = mx + b, then compare the slopes Same slope,different b PARALLEL y = 3x − 4y = 3x + 6 b is −4 and 6 Same slope,same b SAME LINE y = 2x + 3 and4x − 2y = −6 (solved: y = 2x + 3) Slopes multiplyto −1 PERPENDICULAR y = (2/5)x + 1y = −(5/2)x − 3 (2/5)(−5/2) = −1 Anythingelse NEITHER y = 4x + 2y = −4x + 2 product is −16 Vertical lines (x = a) have an undefined slope, so check for them first: two different vertical lines are parallel; a vertical line and a horizontal line are perpendicular.
Solve for y, compare the slopes, and read off the answer. Check for vertical lines first.

The same idea, the fun way

Pick the one that clicks. Each one explains the same concept from a different angle.

Sports

Lines on a rectangular court

On a rectangular court, the two sidelines never meet, and each end line meets them in a square corner. Put the court on a graph with the sidelines along y = 0 and y = 5 and the end lines along x = 0 and x = 8 (made-up units). The two sidelines have the same slope, 0, and different y-intercepts, so they are parallel. The two end lines are both vertical, so they are parallel too. Each end line is vertical and each sideline is horizontal, so they are perpendicular.

How it connects: Horizontal with vertical is the special perpendicular pair, where the flip-and-change-sign rule is not used (0 has no reciprocal). The limit: turn the court on the graph and the sidelines would share some slope m while the end lines would have slope −1/m. And the right angle only looks square if both axes use the same scale.

Gaming

The chess bishop

In chess, a bishop moves along diagonals. Put the board on a graph with square cells. One kind of diagonal goes up and to the right, 1 up for every 1 right, so its slope is 1. The other kind goes up and to the left, so its slope is −1. Two different diagonals of the same kind are parallel, like y = x and y = x + 2. A diagonal of one kind crosses a diagonal of the other kind at a right angle.

How it connects: Slopes 1 and −1 multiply to −1: the reciprocal of 1 is 1, and changing the sign gives −1. The limit: this pair is the easy case. For other slopes, like 3/4, you really need both the flip and the sign change. And it works because the board squares are equal, so the graph has the same scale on both axes.

Relationships

Walking at the same pace

Ava and Ben walk along the same straight road, in the same direction, at the same steady speed: 80 meters per minute. When they start, Ava is already 200 meters farther along the road than Ben. Let t be the minutes since they started and y the position in meters, measured from Ben’s starting point. Then Ben is at y = 80t and Ava is at y = 80t + 200. The gap stays 200 meters, and Ben never catches up.

How it connects: The same slope (the same speed) and different y-intercepts (different starting spots) give parallel lines, and parallel lines never meet. The limit: if Ben speeds up or Ava slows down, the slopes are different and the lines cross, so Ben can catch up. Also, the right angle between lines would mean nothing here, because time and distance use different units.

Proverb

Opposites attract

The saying “opposites attract” means that people who are very different from each other are often drawn together. A perpendicular slope is an “opposite” in two ways at once: the sign is opposite, and the fraction is flipped. The slope 5/9 has the perpendicular partner −9/5. Careful: in math, “opposites” alone means the same number with the other sign, like 5/9 and −5/9, and those slopes are not perpendicular.

How it connects: You need both opposites. Changing only the sign (5/9 and −5/9) does not give perpendicular lines, and neither does flipping only (5/9 and 9/5). The product must be exactly −1: (5/9)(−9/5) = −1. The limit: the saying is about people and is not a rule of nature. For slopes, the rule is exact.

Everyday joke

The two lines that never met

Why did the two parallel lines never become friends? They had so much in common, but they never met.

How it connects: Same slope and different y-intercepts: the lines never meet. The limit: “so much in common” is only the slope. The y-intercepts are different. If they matched too, the two equations would describe the same line, and the lines would meet everywhere.

Memory help

Mnemonic

Parallel has two parallel lines in it

Look at the word: the two ls in the middle of “parallel” are themselves two parallel lines side by side. Parallel lines have the same slope m and different y-intercepts b.

Acronym

SLIP

To write a parallel or perpendicular line through a point: Slope of the given line. Look at what you need (the same slope for parallel, flip and change sign for perpendicular). Insert the point into y = mx + b. Pull out b and write the equation.

Rhyme

The slope rhyme

“Same m but a different b: parallel lines, you see. Flip the slope and change its sign: perpendicular, you’re doing fine.”

Picture it

Tilt the plus sign

Draw a plus sign: one horizontal line and one vertical line, the basic perpendicular pair. Now imagine tilting the whole plus sign. Both slopes change, but the corner stays square, and as long as neither line is horizontal or vertical, the two slopes multiply to −1.

Trap alert

Opposite is not enough

Slopes 4 and −4 are opposites, but 4 × (−4) = −16, so the lines are not perpendicular. Slopes 4 and 1/4 are reciprocals, but 4 × (1/4) = 1, so they are not perpendicular either. You need the flip and the sign change: 4 and −1/4.

Trap alert

Solve for y before you read the slope

In 3x + 4y = 8 the number in front of x is 3, but the slope is −3/4. Solve for y first: y = −(3/4)x + 2.

Study notes

Key ideas

  • Parallel lines are two different lines that never meet. Non-vertical parallel lines have the same slope and different y-intercepts. Two different vertical lines are parallel too.
  • Perpendicular lines cross at a right angle. If neither line is horizontal or vertical, their slopes are negative reciprocals: flip the fraction (the sign stays), then change the sign. The product is −1.
  • A horizontal line (slope 0) and a vertical line (undefined slope) are perpendicular. The reciprocal rule does not apply to them.
  • To test two lines, write both as y = mx + b first, then compare the slopes. Same slope and same y-intercept means the same line.
  • To write a parallel or perpendicular line through a point, get the new slope from the given line, then substitute the point to find b.

Steps

  1. Find the slope of the given line (solve for y if needed). If the line is horizontal or vertical, use the special rules instead.
  2. Parallel: keep the slope. Perpendicular: flip it and change the sign.
  3. Put the new slope and the point (x1, y1) into y = mx + b and solve for b.
  4. Write the equation and check it: the point must fit, and the slopes must match (parallel) or multiply to −1 (perpendicular).

Formulas

  • Parallel: m1 = m2 and b1 ≠ b2 (different lines)
  • Perpendicular: m1 × m2 = −1, so m2 = −1/m1 (when m1 ≠ 0)
  • Slope from standard form: Ax + By = C with B ≠ 0 has slope −A/B and y-intercept C/B
  • Special pair: y = b (horizontal) and x = a (vertical) are perpendicular

Watch out

  • Perpendicular needs both the flip and the sign change: 3/4 goes with −4/3, not with −3/4 or 4/3.
  • Solve for y before reading a slope: in 3x + 4y = 8, the slope is −3/4, not 3.
  • The point’s y-value is not b unless the point’s x is 0.
  • The same slope and the same y-intercept means the same line, not parallel lines.
  • Do not flip 0. A horizontal line’s perpendicular is vertical. And in y = 5 the 5 is the y-intercept, not the slope.

Why it works

  • Why do parallel lines never meet? If y = mx + b1 and y = mx + b2 met, then mx + b1 = mx + b2 there, so b1 = b2. They are different lines, so that cannot happen.
  • Why a negative reciprocal? A quarter turn swaps run and rise and changes one sign, so slope a/b becomes −b/a.
  • Why can’t you flip 0? 1 ÷ 0 is undefined. A horizontal line’s perpendicular is vertical, and a vertical line has an undefined slope.

Common mistakes

✗ MistakePerpendicular lines have opposite slopes, like 3 and −3.
✓ InsteadPerpendicular slopes are negative reciprocals: 3 and −1/3. Check: 3 × (−3) = −9, but 3 × (−1/3) = −1.
✗ MistakePerpendicular lines have reciprocal slopes, like 3 and 1/3.
✓ InsteadYou also need the sign change: 3 and −1/3. Check: 3 × (1/3) = 1, not −1.
✗ MistakeThe slope of 4x + 3y = 12 is 4.
✓ InsteadSolve for y first: 3y = −4x + 12, so y = −(4/3)x + 4. The slope is −4/3.
✗ MistakeTwo lines with the same slope are always parallel.
✓ InsteadIf the y-intercepts are also the same, the two equations describe the same line. Parallel lines have the same slope and different y-intercepts.
✗ MistakeThe line perpendicular to y = 5 is y = −(1/5)x, because the negative reciprocal of 5 is −1/5.
✓ Insteady = 5 is horizontal, with slope 0. The 5 is the y-intercept, not the slope. A perpendicular line is vertical, x = a for some number a.

Teach it back

Explain to a friend how to write the equation of the line through (2, 4) that is perpendicular to y = (1/5)x + 2, and how you can tell your answer is perpendicular.

Explaining it out loud shows what you really know. Did you…

  • I took the given slope, 1/5, flipped it to 5, and changed the sign to get −5.
  • I put m = −5 and the point into y = mx + b: 4 = −5(2) + b, so b = 14.
  • I wrote y = −5x + 14 and checked that it passes through (2, 4): −5(2) + 14 = 4.
  • I checked that the slopes multiply to −1: (1/5)(−5) = −1.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

Downloads a calendar file (.ics). Works with Google, Apple and Outlook calendars. Nothing is sent to us.

Practice it your way

Flashcards

Flip, recall and space your reviews.

Cheat sheet

One page. Everything you need.

True or false

Fast yes/no questions with reasons.

Matching game

Pair them up against the clock.

Practice questions

Step-up questions with explanations.

Slope of a LineSlope-Intercept FormGraphing Linear EquationsPoint-Slope FormStandard Form of a LineSolving Systems by Graphing