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

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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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Standard form vs slope-intercept form at a glance

Both forms describe the same straight line. They just make different facts easy to see. Standard form is handy for finding intercepts. Slope-intercept form shows the slope and the starting point on the y-axis right away.

Quick check for the example: slope = −4/(−2) = 2 and y-intercept = 6/(−2) = −3. That matches y = 2x − 3.

Common mistakes when converting

• Forgetting the sign when moving Ax. In 5x + y = 7, subtracting 5x gives y = −5x + 7, not y = 5x + 7.
• Dividing only one term by B. In 3x + 4y = 12, you must divide both −3x and 12 by 4: y = −(3/4)x + 3.
• Losing a negative B. In 2x − y = 5, you get −y = −2x + 5. Divide everything by −1 to get y = 2x − 5.
• Mixing up the shortcut. The slope is −A/B, not A/B. Plug in one point from your answer to make sure it still satisfies the original equation.

A good habit: after converting, test x = 0. The y-value you get should satisfy the original standard form equation.

Frequently Asked Questions

How do you convert standard form to slope-intercept form?

Solve the equation for y. Subtract the x-term from both sides, then divide every term by the coefficient of y. For 4x + 2y = 10, subtract 4x to get 2y = −4x + 10, then divide by 2: y = −2x + 5. The slope is −2 and the y-intercept is 5.

How do you convert 2x + 3y = 12 into slope-intercept form?

Subtract 2x from both sides to get 3y = −2x + 12. Then divide each term by 3: y = −(2/3)x + 4. The slope is −2/3 and the line crosses the y-axis at (0, 4). Check with x = 3: the original gives 6 + 3y = 12, so y = 2, and −(2/3)(3) + 4 = 2 as well.

How do you convert slope-intercept form to standard form?

Move the x-term to the left side and clear any fractions. For y = −(3/4)x + 2, multiply every term by 4 to get 4y = −3x + 8, then add 3x to both sides: 3x + 4y = 8. Most teachers want A positive and A, B and C written as integers with no common factor.

How do you convert general form to slope-intercept form?

General form is Ax + By + C = 0, with everything on one side. Move the x-term and the constant to the right, then divide by B. For 2x − y + 7 = 0, you get −y = −2x − 7, and dividing by −1 gives y = 2x + 7. Watch the signs when B is negative.

How do you find the slope from a standard form equation?

For Ax + By = C, the slope is −A/B and the y-intercept is C/B, as long as B is not zero. In 5x − 10y = 20, the slope is −5/(−10) = 1/2 and the y-intercept is 20/(−10) = −2, so the line is y = (1/2)x − 2. If B = 0, the line is vertical and its slope is undefined.

What is the difference between slope-intercept form and standard form?

Slope-intercept form, y = mx + b, shows the slope and y-intercept right away, which makes graphing quick. Standard form, Ax + By = C, keeps x and y on one side and makes both intercepts easy to find: in 3x + 4y = 12, set x = 0 to get y = 3 and y = 0 to get x = 4. Both describe the same line.