2 Warm Up Evaluate each expression. 1. 9 –3(–2) 2. 3(–5 + 7) 15 3. 4. 26 – 4(7 – 5)Simplify each expression.5. 10c + c156–41811c
3 ObjectiveSolve two-step and multi-step equations in one variable.
4 Alex belongs to a music club Alex belongs to a music club. In this club, students can buy a student discount card for $ This card allows them to buy CDs for $3.95 each. After one year, Alex has spent $63.40.To find the number of CDs c that Alex bought, you can solve an equation.Many equations contain more than one operation, such as 2x + 5 = 11.This equation contains multiplication and addition. Equations that contain two operations require two steps to solve. Identify the operations in the equation and the order in which they are applied to the variable. Then use inverse operations to undo them in reverse order one at a time.
5 Operations in the Equation 2x + 5 = 11Operations in the EquationTo Solve1. First x is multiplied by 2.1. Subtract 5 from both sides of the equation.2. Then 5 is added.3.95c = 63.402. Then divide both sides by 2.
6 2x + 5 = 11 2x + 5 = 11 Subtract 5 from both sides of the equation. −5 −52x = 6x = 3Divide both sides of the equation by 2.Each time you perform an inverse operation, you create an equation that is equivalent to the original equation. Equivalent equations have the same solutions. In the example above, 2x + 5 = 11, 2x = 6, and x = 3 are all equivalent equations.
7 Additional Example 1A: Solving Two-Step Equations Solve 18 = 4a + 10.18 = 4a + 10First a is multiplied by 4. Then 10 is added. Subtract 10 from both sides.– – 108 = 4aSince a is multiplied by 4, divide both sides by 4 to undo the multiplication.8 = 4a42 = a
8 Check It Out! Example 1aSolve –4 + 7x = 3.–4 + 7x = 3First x is multiplied by 7. Then –4 is added. Add 4 to both sides.7x = 7Since x is multiplied by 7, divide both sides by 7 to undo the multiplication.7x = 77x = 1
9 Check It Out! Example 1cSolveFirst n is divided by 7. Then 2 is added. Subtract 2 from both sides.–2 –2Since n is divided by 7, multiply both sides by 7 to undo the division.n = 0
10 Method 1 Use fraction operations. Additional Example 2A: Solving Two-Step Equations That Contain FractionsSolveMethod 1 Use fraction operations.Since is subtracted from , add to both sides to undo the subtraction.34y8Since y is divided by 8, multiply both sides by 8 to undo the division.
11 Additional Example 2A Continued SolveMethod 1 Use fraction operations.Multiply both sides by 8.Simplify.y = 16
12 Method 1 Use fraction operations. Additional Example 2B: Solving Two-Step Equations That Contain FractionsSolveMethod 1 Use fraction operations.Since is added to r, subtract from both sides to undo the addition.342The reciprocal of is Since r is multiplied by , multiply both sides by .32
13 Additional Example 2B Continued SolveMethod 2 Multiply by the LCD to clear the fractions.Multiply both sides by 12, the LCD of the fractions.Distribute 12 on the left side.8r + 9 = 7Simplify. Since 9 is added to 8r, subtract 9 from both sides to undo the addition.–9 –98r = –2