Solving Two-Step Equations 2-8 Solving Two-Step Equations Course 3 Warm Up Problem of the Day Lesson Presentation

Warm Up Solve. 1. x + 12 = 35 2. 8x = 120 3. = 7 4. –34 = y + 56 x = 23 x = 15 y9 y = 63 y = –90

Problem of the Day x is an odd integer. If you triple x and then subtract 7, you get a prime number. What is the smallest possible x? (Hint: What is the smallest prime number?) x = 3

Learn to solve two-step equations.

Sometimes more than one inverse operation is needed to solve an equation. Before solving, ask yourself, “What is being done to the variable, and in what order?” Then work backward to undo the operations.

Additional Example 1: Problem Solving Application The mechanic’s bill to repair Mr. Wong’s car was $650. The mechanic charges $45 an hour for labor, and the parts that were used cost $443. How many hours did the mechanic work on the car?

Understand the Problem Additional Example 1 Continued 1 Understand the Problem List the important information: The answer is the number of hours the mechanic worked on the car. The parts cost $443. The labor cost $45 per hour. The total bill was $650. Let h represent the hours the mechanic worked. Total bill = Parts + Labor 650 = 443 + 45h

Additional Example 1 Continued 2 Make a Plan Think: First the variable is multiplied by 45, and then 443 is added to the result. Work backward to solve the equation. Undo the operations in reverse order: First subtract 443 from both sides of the equation, and then divide both sides of the new equation by 45.

Additional Example 1 Continued Solve 3 650 = 443 + 45h –443 –443 Subtract to undo the addition. 207 = 45h 207 45h 45 45 = Divide to undo multiplication. 4.6 = h The mechanic worked for 4.6 hours on Mr. Wong’s car.

Additional Example 1 Continued Look Back 4 You can use a table to decide whether your answer is reasonable. Hours Labor Parts Total Cost 1 45 $443 $488 2 90 $533 3 135 $578 4 180 $623 5 225 $668 4.6 hours is a reasonable answer.

Check It Out: Example 1 The mechanic’s bill to repair your car was $850. The mechanic charges $35 an hour for labor, and the parts that were used cost $275. How many hours did the mechanic work on your car?

Understand the Problem Check It Out: Example 1 Continued 1 Understand the Problem List the important information: The answer is the number of hours the mechanic worked on your car. The parts cost $275. The labor cost $35 per hour. The total bill was $850. Let h represent the hours the mechanic worked. Total bill = Parts + Labor 850 = 275 + 35h

Check It Out: Example 1 Continued 2 Make a Plan Think: First the variable is multiplied by 35, and then 275 is added to the result. Work backward to solve the equation. Undo the operations in reverse order: First subtract 275 from both sides of the equation, and then divide both sides of the new equation by 35.

Check It Out: Example 1 Continued Solve 3 850 = 275 + 35h –275 –275 Subtract to undo the addition. 575 = 35h 575 35h 35 35 = Divide to undo multiplication. 16.4  h The mechanic worked for about 16.4 hours on your car.

Check It Out: Example 1 Continued Look Back 4 You can use a table to decide whether your answer is reasonable. Hours Labor Parts Total Cost 13 455 $275 $730 14 490 $765 15 525 $800 16 560 $835 17 595 $870 16.4 hours is a reasonable answer.

Additional Example 2A: Solving Two-Step Equations Solve + 7 = 22 Method 1: Work backward to isolate the variable. Think: First the variable is divided by 3, and then 7 is added. To isolate the variable, subtract 7, and then multiply by 3. + 7 – 7 = 22 – 7 n 3 Subtract 7 from both sides. 3  = 3  15 n3 Multiply both sides by 3. n = 45

Additional Example 2A Continued Solve + 7 = 22 Method 2: Multiply both sides of the equation by the denominator. + 7 = 22(3) n3 (3) Multiply both sides by the denominator. n + 21 = 66 Subtract to undo addition. –21 –21 n = 45

Additional Example 2B: Solving Two-Step Equations y – 4 3 Solve = 9 Method 1: Work backward to isolate the variable. – = 9 y3 43 Rewrite the expression as the sum of two fractions. Think: First the variable is divided by 3, and then is subtracted. To isolate the variable, add and then multiply by 3. 43 Add to both sides. 43 43 – + = 9 + y3 (3) = (3) y3 31 t3 Multiply both sides by 3. y = 31

Additional Example 2B: Solving Two-Step Equations y – 4 3 Solve = 9 Method 2: Multiply both sides of the equation by the denominator. = 9 y – 4 3 = 9 y – 4 3 (3) (3) Multiply both sides by the denominator. y – 4 = 27 + 4 + 4 Add to undo subtraction. y = 31

Check It Out: Example 2A n4 Solve + 8 = 18 Method 1: Work backward to isolate the variable. Think: First the variable is divided by 4, and then 8 is added. To isolate the variable, subtract 8, and then multiply by 4. + 8 – 8 = 18 – 8 n 4 Subtract 8 from both sides. 4  = 4  10 n4 Multiply both sides by 4. n = 40

Check It Out: Example 2A n4 Solve + 8 = 18 Method 2: Multiply both sides of the equation by the denominator. + 8 = 18(4) n4 (4) Multiply both sides by the denominator. n + 32 = 72 Subtract to undo addition. –32 –32 n = 40

Check It Out: Example 2B y – 7 2 Solve = 7 Method 1: Work backward to isolate the variable. – = 7 y2 72 Rewrite the expression as the sum of two fractions. Think: First the variable is divided by 2, and then is subtracted. To isolate the variable, add and then multiply by 2. 72 Add to both sides. 72 72 – + = 7 + y2 (2) = (2) y2 21 t2 Multiply both sides by 2. y = 21

Check It Out: Example 2B y – 7 2 Solve = 7 Method 2: Multiply both sides of the equation by the denominator. = 7 y – 7 2 = 7 y – 7 2 (2) (2) Multiply both sides by the denominator. y – 7 = 14 + 7 + 7 Add to undo subtraction. y = 21

Lesson Quiz Solve. 1. – 3 = 10 2. 7y + 25 = –24 3. –8.3 = –3.5x + 13.4 4. = 3 5. The cost for a new cell phone plan is $39 per month plus a one-time start-up fee of $78. If you are charged $1014, how many months will the contract last? x –9 x = –117 y = –7 x = 6.2 y + 5 11 y = 28 24