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.
Subtract mx from both sides
y - mx = (mx - mx ) + b
y - mx = b
Subtract b from both sides (optional)
y - mx - b = b - b
y - mx - b = 0
Rearrange to put the x term first
-mx + y - b = 0
Let the fraction A/B represent m
If m is an integer, then B will equal 1.
-A/Bx + y - b = 0
Multiply both sides of the equation by the denominator B
-Ax + By - Bb = 0
Let Bb = C
-Ax + By - C = 0
Examples:
(1) - The equation of a line in slope intercept form is y = 1/2 x + 5. What is the equation in standard form?
Subtract 1/2 x from both sides of the equation
y - 1/2x = 5
Subtract 5 from both sides
y - 1/2x - 5 = 0
Multiply both sides by the denominator of the fraction
2y - x - 10 = 0
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 = -3/7x + 10. Following the procedure outlined above:
y + 3/7x - 10 = 0
7y + 3x - 70 = 0
3x + 7y -70 = 0 or 3x + 7y = 70
