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➕ Algebra 1 · Linear Functions

Point-Slope Form: notes and common mistakes

Study notes

Key ideas

  • Point-slope form is y − y1 = m(x − x1): m is the slope and (x1, y1) is any one known point on the line.
  • In y − y1 = m(x − x1), flip the sign you see next to x and y to read the point: x + 3 means x1 = −3, and y − 5 means y1 = 5. The slope keeps its own sign.
  • With two points, find the slope first, then use either point. Both choices describe the same line.
  • A non-vertical line has endless point-slope equations (one for each point on it) but only one slope-intercept equation.
  • To convert to slope-intercept form, distribute the slope and then move the number next to y to the right side.
  • A vertical line has an undefined slope, so it has no point-slope form. Its equation is x = a.

Steps

  1. If you have two points, find m = (y2 − y1) ÷ (x2 − x1). If the run is 0, stop: the line is vertical and its equation is x = a.
  2. Choose one point to be (x1, y1).
  3. Write y − y1 = m(x − x1), putting parentheses around negative numbers, then simplify the double negatives (y − (−2) is y + 2).
  4. If asked, distribute and solve for y to reach slope-intercept form.
  5. Check with the other point (or with the known point).

Formulas

  • Point-slope form: y − y1 = m(x − x1)
  • Slope from two points: m = (y2 − y1) ÷ (x2 − x1), assuming x2 ≠ x1
  • To slope-intercept form: y = mx + b, where b = y1 − m × x1
  • Horizontal line (m = 0): y − y1 = 0, which is y = y1
  • Vertical line: x = a (slope undefined, no point-slope form)

Watch out

  • Read the signs inside as opposites: y + 4 means y1 = −4, not 4.
  • Keep x and y as letters. Only x1, y1 and m are replaced by numbers.
  • With two points, find m first. Do not put both points into the form without a slope.
  • When you distribute, multiply the slope by both terms in the parentheses, including the sign of the second term.
  • A vertical line has an undefined slope, not a slope of 0.

Why it works

  • Why does the known point fit the equation? Put x = x1 in: the right side is m × 0 = 0, so y = y1.
  • Why do the signs flip? The form subtracts the known coordinate, and subtracting a negative number is the same as adding.
  • Why do all the point-slope equations of one line agree? Every pair of points on a line gives the same slope, so each equation says the same thing: rise = slope × run. After simplifying, they all give the same y = mx + b.

Common mistakes

✗ MistakeIn y + 3 = 2(x − 5), the point is (5, 3).
✓ InsteadThe y + 3 is y − (−3), so y1 = −3. The point is (5, −3). Flip the sign you see next to x and y.
✗ MistakeTo write the equation, replace x and y with the coordinates of the point.
✓ Insteadx and y stay as letters, because they stand for every point on the line. Only x1, y1 and m get numbers.
✗ MistakeEach line has exactly one point-slope equation.
✓ InsteadAny point on a non-vertical line can be the known point, so there are endless point-slope equations. They all simplify to the same y = mx + b.
✗ MistakeWith two points, put both into y − y1 = m(x − x1) and you are done.
✓ InsteadThe form needs m. Find the slope with (y2 − y1) ÷ (x2 − x1) first, then use only one point in the form.
✗ MistakeA vertical line has point-slope form with m = 0.
✓ Insteadm = 0 is a horizontal line. A vertical line has an undefined slope, so it has no point-slope form. Its equation is x = a.

Teach it back

Explain to a friend how to write the equation of the line through (−2, 9) and (4, 3) in point-slope form and in slope-intercept form, and why two different-looking point-slope equations can describe the same line.

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

  • I found the slope first: rise 3 − 9 = −6 and run 4 − (−2) = 6, so m = −1.
  • I picked a known point, for example (4, 3), and wrote y − 3 = −1(x − 4).
  • I distributed and added 3 to get y = −x + 7, then checked it with the other point: −(−2) + 7 = 9.
  • I explained that the other point (−2, 9) gives y − 9 = −1(x + 2), which simplifies to the same y = −x + 7.

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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