Convert Standard Form to Slope-Intercept Form: Steps and Examples

To convert Ax + By = C to slope-intercept form, subtract Ax from both sides and divide every term by B: y = (−A/B)x + C/B, provided B ≠ 0. The slope is −A/B, and the y-intercept is (0, C/B). If B = 0, the line is vertical and has no slope-intercept form.

How to convert standard form to slope-intercept form

Standard form writes a line as Ax + By = C, with A and B not both zero. Many classes use integer coefficients. Slope-intercept form writes a nonvertical line as y = mx + b, where m is the slope and (0, b) is the y-intercept.

1. Move the x-term: subtract Ax from both sides to get By = −Ax + C.
2. Isolate y: divide both sides by B. This gives y = (−Ax + C)/B = (−A/B)x + C/B.
3. Simplify both fractions: compare your result with y = mx + b to identify m and b.

Keep the sign of B. If B is negative, divide both terms on the right by that negative number. Dividing only the x-term changes the equation.

What if the equation is Ax + By + C = 0?

This convention puts the constant on the left. Move it to the right first: Ax + By = −C. For B ≠ 0, the result is y = (−A/B)x − C/B. The two conventions describe lines in different arrangements; use the constant with the sign it has after moving it to the right.

Worked conversion examples

Example 1: a negative coefficient of y

Convert x − 2y = −6.

1. Subtract x: −2y = −x − 6.
2. Divide every term by −2: y = (1/2)x + 3.

The slope is 1/2, and the y-intercept is (0, 3). As a quick check, x = 0 gives y = 3 in both equations: 0 − 2(3) = −6.

Example 2: a negative slope

Convert 5x + 4y = 12. Subtract 5x to get 4y = −5x + 12. Divide by 4: y = −(5/4)x + 3. The slope is −5/4, and the y-intercept is (0, 3). Keep the exact fraction rather than rounding it.

Example 3: all terms on the left

Convert 2x + 3y + 6 = 0. Move 2x and 6 to the right: 3y = −2x − 6. Divide by 3: y = −(2/3)x − 2. Here b is −2, not +2.

Horizontal and vertical lines

• A = 0, B ≠ 0: By = C becomes y = C/B. It is a horizontal line with slope 0. For example, 4y = 12 gives y = 3.
• B = 0, A ≠ 0: Ax = C becomes x = C/A. It is a vertical line with undefined slope. For example, 3x = 9 gives x = 3; do not divide by B or try to write y = mx + b.
• A = B = 0: the equation is not a line. If C = 0, every point satisfies it; if C ≠ 0, no point satisfies it.

Practice: convert each equation

Work on paper first. For each nonvertical line, give the equation in slope-intercept form, its slope, and its y-intercept.

1. 7x − 2y = 5
2. 2x − 6y = −11
3. 3x − 3y = 12
4. 12x − 12y = 5
5. 5y = −10
6. 2x = −8

Check your conversion

Multiply your final equation by the original coefficient of y and rearrange it. You should recover the original equation. Substituting a convenient point can catch a sign error, but checking just one point does not prove that two equations are equivalent.

If your next step is drawing the line, use graphing linear equations. To reverse the conversion, review writing a line in standard form. For more algebra practice, choose a topic in the Algebra 1 learning center.

Reference: OpenStax, slope-intercept form. The examples and practice above are explained here for this lesson.

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