The standard form of a straight line is:
Ax + By = C
The usual rules are that A, B and C are whole numbers, A is not negative, and the three numbers share no common factor. So 2x + 3y = 12 is in standard form, but 4x + 6y = 24 should be divided by 2 first.
Converting from y = mx + b
Example 1
Write y = −⅔x + 4 in standard form.
Multiply every term by 3 to clear the fraction: 3y = −2x + 12.
Move the x term to the left: 2x + 3y = 12.
Example 2
Write y = ¾x + ½ in standard form.
Multiply by 4, the lowest common denominator: 4y = 3x + 2.
Move the x term: −3x + 4y = 2. A must not be negative, so multiply everything by −1: 3x − 4y = −2.
Check with the point (2, 2): 3 × 2 − 4 × 2 = 6 − 8 = −2. ✓
Straight from two points
For points (x₁, y₁) and (x₂, y₂), you can skip the slope altogether:
- A = y₂ − y₁
- B = x₁ − x₂
- C = A·x₁ + B·y₁
For (−2, −1) and (2, 2): A = 3, B = −4, and C = 3 × (−2) + (−4) × (−1) = −6 + 4 = −2. That gives 3x − 4y = −2, the same as Example 2. This method also works for vertical lines, which slope-intercept form can’t handle.
See this line in Slope Finder →
Reading standard form
- Slope: m = −A ÷ B. For 2x + 3y = 12, m = −2/3.
- x-intercept: set y = 0. For 2x + 3y = 12, 2x = 12, so x = 6.
- y-intercept: set x = 0. For 2x + 3y = 12, 3y = 12, so y = 4.
The two intercepts give a fast way to graph: plot (6, 0) and (0, 4) and join them.
Horizontal and vertical lines
- x = 5 is standard form with A = 1, B = 0 and C = 5. It is a vertical line.
- y = −2 is standard form with A = 0, B = 1 and C = −2. It is a horizontal line.