Solving Linear Equations and Inequalities 159
Solving Two-Variable Linear Equations for a Specifi c Variable
You can use the procedures for solving a linear equation in one variable x
to solve a two-variable linear equation, such as 6x + 2 y = 10, for one of the
variables in terms of the other variable. As you solve for the variable of inter-
est, you simply treat the other variable as you would a constant. Often, you
need to solve for y to facilitate the graphing of an equation. (See Chapter 17
for a fuller discussion of this topic.) Here is an example.
Problem Solve 6x + 2 y = 10 for y.
Solution
Step 1. 6 x is added to the variable term 2y, so sub-
tract 6x from both sides.
6 x+ 2 y − 6 x= 10 − 6 x
Step 2. Simplify.
2 y = 10 − 6x
Step 3. You want the coeffi cient of y to be 1, so divide both sides by 2.
2 y 10 6 x
22
=
−
Step 4. Simplify.
y= 5 − 3x
Solving Linear Inequalities
If you replace the equal sign in a linear equation with <, >, ≤, or ≥, the
result is a linear inequality. You solve linear inequalities just about the same
way you solve equations. There is just one important difference. When you
multiply or divide both sides of an inequality by a negative number, you
must reverse the direction of the inequality symbol. To help you understand
why you must do this, consider the two numbers, 8 and 2. You know that
8 > 2 is a true inequality because 8 is to the right of 2 on the number line, as
shown in Figure 14.1.
If you multiply both sides of the inequality 8 > 2 by a negative number,
say, −1, you must reverse the direction of the inequality so that you will still
When you are solving 6x + 2 y =
10 for y, treat 6x as if it were a
constant.
10 6
2
− x ≠ 5−6x. You must divide
both terms of the numerator by 2.
