Skill: line-forms

Point-Slope & Standard Forms

Slope-intercept is not the only way to name a line. Point-slope form is built for “point plus slope” problems, and standard form Ax + By = C hands you both intercepts for free. Learn to convert fluently between all three.

1 Three names for one line

The same line can be written three ways. Each form is the fastest starting point for a different job.

The three forms

y − y₁ = m(x − x₁) · y = mx + b · Ax + By = C

Point-slope when you know a point and the slope. Slope-intercept when you want to graph fast. Standard (integers, A > 0) when you want intercepts.

One line, three outfits: y − 2 = 3(x − 1), y = 3x − 1, 3x − y = 1.

Conversions: point-slope → slope-intercept by distributing and solving for y. Slope-intercept → standard by clearing denominators and moving x left (keep A positive). Standard → intercepts by plugging in 0: x-intercept is C/A, y-intercept is C/B.

2 Worked examples

Write it, convert it, read intercepts off it.

Example 1 Point-slope from point and slope

Through (1, 2) with slope 3:

y − 2 = 3(x − 1) — built for exactly this job.

Example 2 Signs in point-slope

Through (−2, −5) with slope 2:

y + 5 = 2(x + 2) — subtracting a negative becomes plus. The #1 sign trap.

Example 3 Standard to slope-intercept

Convert 2x + 3y = 6:

3y = −2x + 6, so y = −(2/3)x + 2.

Example 4 Intercepts from standard form

Find the intercepts of 2x − 4y = 8:

x-intercept: set y = 0, 2x = 8, so x = 4. y-intercept: set x = 0, −4y = 8, so y = −2.

3 Common mistakes

Three traps, each with the wrong version and the fix.

1. Sign errors with point-slope

Wrong: through (−2, −5), slope 2: y − 5 = 2(x − 2). Right: y − (−5) = 2(x − (−2)), i.e. y + 5 = 2(x + 2).

2. Plugging into y = mx + b instead of using point-slope

Wrong: solving for b first every time — slow and error-prone. Right: point-slope is built for point-plus-slope: y − y₁ = m(x − x₁).

3. Leaving standard form with a negative A

Wrong: −2x + 3y = 6 as a final answer. Right: multiply by −1: 2x − 3y = −6 (A > 0 by convention).

4 Quick checks

Try these yourself, then reveal the answer.

Write point-slope form through (4, −1) with slope 5.

y + 1 = 5(x − 4).

Convert y = 2x − 5 to standard form.

2x − y = 5: move 2x left, keep A positive — 2x − y = 5.

Find the intercepts of 3x + 2y = 12.

x-intercept 4 (12/3); y-intercept 6 (12/2).

5 Key points

Remember

y − y₁ = m(x − x₁) — the point-slope template; watch the signs.
Distribute and solve for y to reach slope-intercept form.
Ax + By = C with integers and A > 0 is standard form.
From standard: x-intercept = C/A, y-intercept = C/B.
Any point on the line works in point-slope — pick the friendliest one.

Key vocabulary

Point-slope form

y − y1 = m(x − x1): built from one point and the slope.

Standard form

Ax + By = C with integer A, B, C and A > 0.

Equivalent forms

Different-looking equations that describe the same line.

Distribute

Multiply m through (x − x1) when leaving point-slope form.

Clear denominators

Multiply through by the LCD so standard form has integers.

Practice conversions Open the form converter Explore in the lab