50 CHAPTER 2 Linear Equations, Inequalities, and Applications
CHECK Original equation
Let
Multiply.- 11 =- 11 ✓ True
- 12 + 6 - 5 4 - 18 + 3
41 - 32 - 21 - 32 - 5 4 + 61 - 32 + 3 x=-3.
4 x- 2 x- 5 = 4 + 6 x+ 3
Use parentheses
around substituted
values to avoid
errors.This is notthe
solution.The true statement indicates that 5 - 36 is the solution set. NOW TRY
CAUTION In Example 2,the equality symbols are aligned in a column. Use
only one equality symbol in a horizontal line of work when solving an equation.
Solving a Linear Equation in One Variable
Step 1 Clear fractions or decimals.Eliminate fractions by multiplying
each side by the least common denominator. Eliminate decimals by
multiplying by a power of 10.
Step 2 Simplify each side separately.Use the distributive property to
clear parentheses and combine like terms as needed.
Step 3 Isolate the variable terms on one side.Use the addition property
to get all terms with variables on one side of the equation and all
numbers on the other.
Step 4 Isolate the variable. Use the multiplication property to get an
equation with just the variable (with coefficient 1) on one side.
Step 5 Check.Substitute the proposed solution into the original equation.OBJECTIVE 4 Solve linear equations by using the distributive property.
In Example 2,we did not use Step 1 or the distributive property in Step 2 as given in
the box. Many equations, however, will require one or both of these steps.
Using the Distributive Property to Solve a Linear EquationSolve
Step 1 Since there are no fractions in this equation, Step 1 does not apply.
Step 2 Use the distributive property to simplify and combine like terms on the left.
Distributive property
Multiply.5 x- 10 =x+ 6 Combine like terms.
2 x- 10 + 3 x=x+ 6
2 x+ 21 - 52 + 3 x=x+ 6
21 x- 52 + 3 x=x+ 6
21 x- 52 + 3 x= x+6.
EXAMPLE 3
NOW TRY ANSWER
- 5 - 46
Be sure to distribute
over allterms within
the parentheses.Step 3 Next, use the addition property of equality.
Add 10.
Combine like terms.
Subtract x.4 x= 16 Combine like terms.
5 x-x= x+ 16 - x
5 x= x+ 16
5 x- 10 + 10 = x+ 6 + 10
NOW TRY
EXERCISE 2
Solve.
5 x+ 11 = 2 x- 13 - 3 x