# How to Convert Slope Intercept Form to Standard Form

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Any equation that relates the first power of x to the first power of y produces a straight line on an x-y graph. The standard form of such an equation is Ax + By + C = 0 or Ax + By = C. When you rearrange this equation to get y by itself on the left side, it takes the form y = mx +b. This is called slope intercept form because m is equal to the slope of the line, and b is the value of y when x = 0, which makes it the y-intercept. Converting from slope intercept form to standard form takes little more than basic arithmetic.

#### TL;DR (Too Long; Didn't Read)

To convert from slope intercept form y = mx + b to standard form Ax + By + C = 0, let m = A/B, collect all terms on the left side of the equation and multiply by the denominator B to get rid of the fraction.

## The General Procedure

An equation in slope intercept form has the basic structure

y = mx + b
1. ## Subtract mx From Both Sides

2. \begin{aligned} y - mx &= (mx - mx ) + b \\ y - mx &= b \end{aligned}
3. ## Subtract b From Both Sides (Optional)

4. \begin{aligned} y - mx - b &= b - b \\ y - mx - b &= 0 \end{aligned}
5. ## Rearrange to Put the x Term First

6. -mx + y - b = 0
7. ## Let the Fraction A/B Represent m

8. If m is an integer, then B will equal 1.

-\frac{A}{B}x + y - b = 0
9. ## Multiply Both Sides of the Equation by the Denominator B

10. -Ax + By - Bb = 0
11. ## Let Bb = C

12. -Ax + By - C = 0

## Examples:

(1) - The equation of a line in slope intercept form is:

y = \frac{1}{2} x + 5

What is the equation in standard form?

1. ## Subtract 1/2 x From Both Sides of the Equation

2. y - \frac{1}{2}x = 5
3. ## Subtract 5 From Both Sides

4. y - \frac{1}{2}x - 5 = 0
5. ## Multiply Both Sides by the Denominator of the Fraction

6. 2y - x - 10 = 0
7. ## Rearrange to Put x as the First Term

8. -x + 2y - 10 = 0

You can leave the equation like this, but if you prefer to make x positive, multiply both sides by -1:

x - 2y + 10 = 0

or

x - 2y = -10

(2) - The slope of a line is -3/7 and the y-intercept is 10. What is the equation of the line in standard form?

The slope intercept form of the line is

y = -\frac{3}{7}x + 10

Following the procedure outlined above:

\begin{aligned} y + \frac{3}{7}x - 10 = 0 \\ 7y + 3x - 70 = 0 \\ 3x + 7y -70 = 0 \\ \text{or} \\ 3x + 7y = 70 \end{aligned}