Converting Slope-Intercept Form to Standard Form

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

(begin{aligned}y – mx &= (mx – mx ) + b y – mx &= bend{aligned})

2. Subtract b From Both Sides (Optional)

(begin{aligned}y – mx – b &= b – b)(y – mx – b &= 0end{aligned})

3. Rearrange to Put the x Term First

(-mx + y – b = 0)

4. Let the Fraction A/B Represent m

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

(-frac{A}{B}x + y – b = 0)

5. Multiply Both Sides of the Equation by the Denominator B

(-Ax + By – Bb = 0)

6. Let Bb = C

(-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

(y – frac{1}{2}x = 5)

2. Subtract 5 From Both Sides

(y – frac{1}{2}x – 5 = 0)

3. Multiply Both Sides by the Denominator of the Fraction

(2y – x – 10 = 0)

4. Rearrange to Put x as the First Term

(-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 = 70end{aligned})