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➕ Algebra 1 · Linear Functions

Parallel and Perpendicular Lines: notes and common mistakes

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.

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