Linear Functions • Topic 5 of 7

Point-Slope Form

Point-slope form, y − y₁ = m(x − x₁), writes a line directly from a slope m and any point (x₁, y₁) — no need to find b first. It is the fastest way to start when you are given a point and a slope, or two points (compute the slope, then use either point). You can leave the answer in point-slope form or expand it to slope-intercept form by distributing and solving for y. The SAT accepts equivalent forms, so point-slope is a handy, low-error starting point for line equations.

✅ Solved examples

1. Write point-slope form for slope 2 through (3, 5).
y − 5 = 2(x − 3).
2. Convert y − 4 = 3(x − 1) to slope-intercept form.
Distribute: y − 4 = 3x − 3; add 4: y = 3x + 1.
3. Write point-slope form for slope −1 through (2, 7).
y − 7 = −1(x − 2), i.e. y − 7 = −(x − 2).
4. A line through (1, 2) and (3, 8): give point-slope form.
Slope = (8 − 2)/(3 − 1) = 3; using (1, 2): y − 2 = 3(x − 1).

✏️ Practice — try these, take hints as needed

1. Write point-slope form for slope 4 through (2, 3).
Use y − y₁ = m(x − x₁).
m = 4, point (2, 3).
—
y − 3 = 4(x − 2).
2. Convert y − 1 = 2(x − 3) to slope-intercept form.
Distribute: y − 1 = 2x − 6.
Add 1.
—
y = 2x − 5.
3. Write point-slope form for slope −3 through (0, 4).
y − 4 = −3(x − 0).
Simplify the bracket.
—
y − 4 = −3x.
4. A line through (2, 1) and (4, 7): slope first, then point-slope.
Slope = (7 − 1)/(4 − 2) = 3.
Use point (2, 1).
—
y − 1 = 3(x − 2).
5. Convert y − 5 = (x − 2) to slope-intercept form.
Distribute (slope 1).
y − 5 = x − 2.
Add 5.
y = x + 3.

📝 Topic test — 8 questions

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