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.

Solution EXAMPLE 4a Solve: 3(k + 1) – 5k = 0

Solution EXAMPLE 4b Solve: 10a – 2(2a – 7) = 68

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 Solve: (1/6)x + 5/2 = 1/3 Solution

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)