We're going to learn how to transform an equation or inequality into an equivalent equation or inequality. This involves rearranging the values of the inequality or equation by using inverse operations. Let's investigate different ways to transform linear equations from one representation to another.
The most common transformation of a linear equation that you will need to know is how to take an equation in standard form (Ax + By = C) and rewrite it in slopeintercept form (y = mx + b), or vice versa. This transformation is important because the two different forms reveal different types of information quickly.
The table below shows the important information in each form.
Form of Linear Equation 
Important Information 
Where You Will See This Form 
SlopeIntercept Form
y = mx + b

The slope is m.
The ycoordinate of the yintercept is b.

Problems that include a starting point (b) and a rate of change.

Standard Form
Ax + By = C

The xcoordinate of the xintercept is C/A.
The ycoordinate of the yintercept is C/B.

Problems that include a combination of multiples of x and y.

To transform from standard form to slopeintercept form usually requires two steps.
Step 1. Add or subtract the x term from both sides.
Step 2. Divide all terms by the coefficient of y.
Example 1
Transform the equation into slopeintercept form: $2y+3x=12$
$2y+3x=12\phantom{\rule{0ex}{0ex}}3x3x\phantom{\rule{0ex}{0ex}}2y=3x+12\phantom{\rule{0ex}{0ex}}\frac{2y}{2}=\frac{3x}{2}+\frac{12}{2}\phantom{\rule{0ex}{0ex}}y=\frac{3}{2}x+6$
To transform from slopeintercept form to standard form usually requires no more than four steps.
Step 1. Add or subtract the x term from both sides.
Step 2. If the coefficient of x is negative (the A term), multiply all terms by 1.
Step 3. If there is a fraction, multiply all terms by the denominator to eliminate fractions.
Step 4. If there is a decimal, multiply all terms by a power of 10 to eliminate decimals.
Example 2
Transform the equation into standard form: $y=\frac{3}{4}x9$
${}_{{{{}_{{{{}_{}}_{}}_{}}}_{}}_{}}y=\frac{3}{4}x9\phantom{\rule{0ex}{0ex}}\frac{3}{4}x\frac{3}{4}x\phantom{\rule{0ex}{0ex}}\frac{3}{4}x+y=9\phantom{\rule{0ex}{0ex}}1(\frac{3}{4}x+y=9)\phantom{\rule{0ex}{0ex}}\frac{3}{4}xy=9\phantom{\rule{0ex}{0ex}}4(\frac{3}{4}xy=9)\phantom{\rule{0ex}{0ex}}3x4y=36$
Check your understanding by completing the following prompts.