Point-slope and standard forms

Three forms of a line

All three describe the same line; you can convert between them.

Point-slope form

y − y₁ = m(x − x₁)

It comes straight from the slope formula: m = (y − y₁)/(x − x₁).

Example: Slope 3 through (2, −1)

y − (−1) = 3(x − 2)

y + 1 = 3(x − 2)

From two points

Example: Through (1, 4) and (3, 10)

• m = (10 − 4)/(3 − 1) = 3
• y − 4 = 3(x − 1)

Converting to slope-intercept

y + 1 = 3(x − 2) → y + 1 = 3x − 6 → y = 3x − 7

Standard form

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

Convert y = (2/3)x − 4 to standard form:

• Multiply by 3: 3y = 2x − 12
• Rearrange: −2x + 3y = −12
• Make A positive: 2x − 3y = 12

Intercepts from standard form

2x − 3y = 12

• x-intercept (y = 0): x = 6 → (6, 0)
• y-intercept (x = 0): y = −4 → (0, −4)

Slope from standard form: m = −A/B = −2/(−3) = 2/3.

Special lines

Choosing a form

• Need to graph quickly from the y-intercept? → slope-intercept.
• Given a point and slope? → point-slope.
• Comparing integer coefficients or finding intercepts? → standard.

Worked examples

Example 1: Line with slope −2 through (−3, 5) in point-slope form: y − 5 = −2(x + 3).

Example 2: Convert y − 5 = −2(x + 3) to standard form: y − 5 = −2x − 6 → 2x + y = −1.

Example 3: Slope of 4x + 5y = 20? −4/5.

Common mistakes

• Sign errors with negative coordinates: through (−3, 5) gives (x + 3), not (x − 3).
• Leaving fractions in standard form.
• Slope from standard form as A/B instead of −A/B.

Try it yourself

1. Write in point-slope form: m = 4, through (1, −2).
2. Convert y = −½x + 3 to standard form.
3. Line through (2, 7) and (6, −1) in slope-intercept form.
4. Intercepts of 5x + 2y = 10.

Answers: 1) y + 2 = 4(x − 1) 2) x + 2y = 6 3) y = −2x + 11 4) (2, 0) and (0, 5)

Quick recap

• Point-slope form: y − y₁ = m(x − x₁).
• Standard form: Ax + By = C; slope = −A/B.
• Convert between forms to suit the task.