Lines and Slope

Point-Slope Form

Write the equation of a line using the point-slope form y - y₁ = m(x - x₁), then convert to slope-intercept form. Equivalence is verified by checking that the original point satisfies the final line equation.

Example

Use a point and slope to write point-slope form, then simplify.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

m=2, (1,5)m= 2 ,\ ( 1 , 5 )
Use slope m = 2 and point (1, 5).(1, 5)m = 2givenm = 2 point (1, 5)y - y0 = m(x - x0)distribute the 2result--

Step 2 — Point-slope form

Use point-slope form with m=2, x-value 1, y-value 5.

y−5=2(x−1)y- \hl{5} = \hl{2} (x- \hl{1} )
Place the point into point-slope form.(1, 5)m = 2givenm = 2 point (1, 5)y - 5 = 2(x - 1)distribute the 2result--

Step 3 — Expand

Distribute: 2 x 1 = 2.

y−5=2x−2y- 5 = 2x - \hl{2}
Distribute: 2 * 1 = 2.(1, 5)m = 2givenm = 2 point (1, 5)y - 5 = 2(x - 1)2 * 1 = 2 y - 5 = 2x - 2result--

Step 4 — Solve for y

Solve for y: y = 2x + 3.

y=2x+3y= 2x + \hl{3}
Solve for y: y = 2x + 3.(1, 5)m = 2givenm = 2 point (1, 5)y - 5 = 2(x - 1)2 * 1 = 2 y - 5 = 2x - 2resultadd 5: -2 + 5 = 3 y = 2x + 3
point-slope-form Given slope m and point (x₁, y₁), write y - y₁ = m(x - x₁). Distribute: y - y₁ = mx - mx₁. Add y₁: y = mx + (y₁ - mx₁). The y-intercept is b = y₁ - mx₁.