Download Simplify Expressions to Solve Equations.

Section 2.2
More about Solving
Equations
Objectives
Use more than one property of equality to
solve equations.
 Simplify expressions to solve equations.
 Clear equations of fractions and decimals.
 Identify identities and contradictions.

Objective 1: Use More Than One Property
of Equality to Solve Equations.

Sometimes we must use several properties of equality to
solve an equation.

For example, on the left side of 2x + 6 = 10, the variable x is
multiplied by 2, and then 6 is added to that product. To isolate
x, we use the order of operations rules in reverse. First, we
undo the addition of 6, and then we undo the multiplication by
2.

The solution is 2.
EXAMPLE 1 Solve: –12x + 5 = 17
Objective 2: Simplify Expressions to Solve
Equations.

When solving equations, we should
simplify the expressions that make up the
left and right sides before applying any
properties of equality.
 Often,
that involves using the distributive
property to removing parentheses and/or
combining like terms.
EXAMPLE 4a Solve: 3(k + 1) – 5k = 0
Solution
EXAMPLE 4b Solve: 10a – 2(2a – 7) = 68
Solution
Objective 3: Clear Equations of Fractions
and Decimals.
Equations are usually easier to solve if
they don’t involve fractions.
 We can use the multiplication property of
equality to clear an equation of fractions
by multiplying both sides of the equation
by the least common denominator of all
the fractions that appear in the equation.

EXAMPLE 6
Solve: (1/6)x + 5/2 = 1/3
Strategy To clear the equations of
fractions, we will multiply both sides by
their LCD of all the fractions in the
equation.
EXAMPLE 6
Solution
Solve: (1/6)x + 5/2 = 1/3
Objective 3: Clear Equations of Fractions
and Decimals

Strategy for Solving Linear Equations in One Variable

Clear the equation of fractions or decimals: Multiply both sides by the LCD to clear
fractions or multiply both sides by a power of 10 to clear decimals.

Simplify each side of the equation: Use the distributive property to remove parentheses,
and then combine like terms on each side.

Isolate the variable term on one side: Add (or subtract) to get the variable term on
one side of the equation and a number on the other using the addition (or subtraction)
property of equality.

Isolate the variable: Multiply (or divide) to isolate the variable using the multiplication (or
division) property of equality.

Check the result: Substitute the possible solution for the variable in the original equation
to see if a true statement results.
Objective 4: Identify Identities and
Contradictions



Each of the equations that we solved in
Examples 1 through 8 had exactly one solution.
Some equations are made true by any
permissible replacement value for the variable.
Such equations are called identities. An
example of an identity is
Objective 4: Identify Identities and
Contradictions



Since we can replace x with any number and the
equation will be true, all real numbers are
solutions of x + x = 2x.This equation has infinitely
many solutions.
Another type of equation, called a contradiction,
is false for all replacement values for the variable.
An example is
Since this equation is false for any value of x, it
has no solution.
EXAMPLE 9
Solve: 3(x + 8) + 5x = 2(12 + 4x)