How To Convert 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.

An equation in slope intercept form has the basic structure

\(y = mx + b\)

1. Subtract mx From Both Sides

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

2. Subtract b From Both Sides (Optional)

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

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

\(-Ax + By – C = 0\)

(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:

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