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➕ Algebra 1 · Linear Functions

Slope-Intercept Form: notes and common mistakes

Study notes

Key ideas

  • Slope-intercept form is y = mx + b: m is the slope and b is the y-intercept, so the line crosses the y-axis at (0, b).
  • In a story, m is the rate of change (the “per” number) and b is the starting value, the value of y when x = 0.
  • To write an equation from a slope and a point, or from two points, find m first, then substitute a point to solve for b.
  • A vertical line cannot be written in slope-intercept form. It is x = a, and its slope is undefined.
  • An equation with a y term, like 3x + 6y = 18, can be rearranged into slope-intercept form by solving for y.

Steps

  1. If you have two points, find the slope: m = (y2 − y1) ÷ (x2 − x1). If the run is 0, stop: the line is vertical and its equation is x = a.
  2. Write y = mx + b with your m in place.
  3. Substitute the x and y of one point and solve for b.
  4. Write the final equation and check it with the other point.

Formulas

  • Slope-intercept form: y = mx + b
  • Finding b from a point: b = y1 − m × x1, using any point (x1, y1) on the line
  • Horizontal line: y = b, slope 0
  • Vertical line: x = a, slope undefined, not in slope-intercept form

Watch out

  • Signs belong to their numbers: in y = 5x − 7, b = −7, not 7.
  • Put the equation in the order y = mx + b first. In y = 4 − 3x, the slope is −3.
  • Do not swap them: m goes with x, and b is alone. In y = 4x + 9, the slope is 4 and the y-intercept is 9, so the line crosses the y-axis at (0, 9).
  • When you divide to solve for y, divide every term.
  • A point’s y-value is not b unless its x is 0.

Why it works

  • Why does b sit at (0, b)? Setting x = 0 makes mx = 0, which leaves y = b.
  • Why is y = mx + b the slope formula in disguise? Use the points (0, b) and (x, y) in m = (y − b) ÷ x, then solve for y.
  • Why can’t a vertical line fit? Its run is 0, so its slope is undefined: there is no number to put in for m.

Common mistakes

✗ MistakeIn y = 4x + 9, the slope is 9 and the y-intercept is 4.
✓ InsteadThe number multiplying x is the slope: m = 4. The number added on its own is the y-intercept: b = 9.
✗ MistakeA line with slope 2 through (5, 14) is y = 2x + 14.
✓ InsteadThe 14 is a y-value at x = 5, not at x = 0. Substitute: 14 = 2(5) + b, so b = 4 and y = 2x + 4.
✗ MistakeEvery straight line can be written as y = mx + b.
✓ InsteadVertical lines (x = a) cannot, because their slope is undefined.
✗ MistakeIn y = 5 − 2x, the slope is 5.
✓ InsteadRewrite it as y = −2x + 5. The slope is −2, and 5 is the y-intercept.

Teach it back

Explain to a friend how to write the equation of the line through (1, 2) and (4, 11), and what m and b tell you about the line.

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

  • I found the slope: rise 11 − 2 = 9 over run 4 − 1 = 3, so m = 3.
  • I put m and one point into y = mx + b and solved for b to get b = −1.
  • I wrote y = 3x − 1 and checked it with the other point.
  • I said what m and b mean: up 3 for each 1 step right, and the line crosses the y-axis at (0, −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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