10-3 Solving Two-Step Equations Problem of the Day Janet used a pedometer to count her steps. The first day and third day she took exactly the same number of steps. The second day she took 1,739 steps. If she took 6,125 steps in all those 3 days, how many steps did she take the first day? 2,193 steps 10-3 Solving Two-Step Equations Solving Two-Step Equations 10-3 Solving Two-Step Equations Solving Two-Step Equations MA.6.A.3.3 Works backward with two-step functions rules to undo expressions. Also MA.6.A.3.2 10-3 Solving Two-Step Equations Solving Two-Step Equations Essential Question: How are equations applied in the real world? 10-3 Solving Two-Step Equations Warm Up – Refresh your memory: Solve each equation. 1. 15x = 225 x = 15 2. y + 2 = 10 y=8 Find the value of each expression. 3. 18 – 6 3 4. 7 + 3 5 8 (16 4) – 2 17 10-3 Solving Two-Step Equations Solving Two-Step Equations Solve each equation. 18 + 3x 18 + 3x – 18 3x = 30 = 30 –18 = 12 Subtract 18 from both sides to undo the addition. 3x = 12 3 3 Divide both sides by 3 to undo the multiplication. x=4 10-3 Solving Two-Step Equations Check 18 + 3x = 30 ? 30 18 + 3(4)= Substitute 4 for x in the equation. ? 30 18 +12 = ? 30 30 = 4 is the solution. 10-3 Solving Two-Step Equations Helpful Hint Undo operations in the reverse of the Order of Operations. First undo addition or subtraction. Then undo multiplication or division. 10-3 Solving Two-Step Equations Solving Two-Step Equations Solve. Check the answer. x –2= 1 3 x –2= 1 3 + 2 +2 x = 3 3 3· x= 3·3 3 x=9 Add 2 to both sides to undo the subtraction. Multiply both sides by 3. 10-3 Solving Two-Step Equations Check x –2 = 1 3 9 ? 1 –2 = 3 ? = 3 –2 1 ? 1 = 1 Substitute 9 for x in the equation. 9 is the solution. 10-3 Solving Two-Step Equations Check It Out: Example 1A Solve each equation. 12 + 3x = 27 12 + 3x = 27 – 12 –12 3x = 15 Subtract 12 from both sides to undo the addition. 3x = 15 3 3 Divide both sides by 3 to undo the multiplication. x=5 10-3 Solving Two-Step Equations Check It Out: Example 1A Continued Check 12 + 3x = 27 ? 27 12 + 3(5)= Substitute 5 for x in the equation. ? 27 12 +15 = ? 27 27 = 5 is the solution. 10-3 Solving Two-Step Equations Check It Out: Example 1B Solve. Check the answer. x –1= 2 2 x –1= 2 2 + 1 +1 x = 3 2 2· x= 2·3 2 x=6 Add 1 to both sides to undo the subtraction. Multiply both sides by 2. 10-3 Solving Two-Step Equations Check It Out: Example 1B Continued Check x –1 =2 2 6 ? 2 –1 = 2 ? = 3 –1 2 ? 2 = 2 Substitute 6 for x in the equation. 6 is the solution. 10-3 Solving Two-Step Equations Consumer Math Application Nancy saved $87 of the money she made babysitting. She wants to buy CDs that cost $15 each, along with a set of headphones that costs $12. How many CDs can she buy? Write a two-step equation to represent the situation. Let x represent the number of CDs Nancy can buy. cost of a CD times + the number of CDs 15x + cost of = headphones 12 = total cost 87 The equation 15x + 12 = 87 represents the situation. 10-3 Solving Two-Step Equations Nancy saved $87 of the money she made babysitting. She wants to buy CDs that cost $15 each, along with a set of headphones that costs $12. How many CDs can she buy? Solve the equation. 15x + 12 – 12 15x 15x 15 x = 87 –12 = 75 = 75 15 =5 Nancy can buy 5 CDs. Subtract 12 from both sides. Divide both sides by 15. 10-3 Solving Two-Step Equations Check It Out: Example 2B Jack rented a car while they were on vacation. He paid a rental fee of $20.00 per day, plus 20¢ a mile. He paid $25.00 for mileage and his total bill for renting the car was $165.00. For how many days did he rent the car? Write a two-step equation to represent the situation. Let d represent the number of days he rented the car. cost of rental fee + times number of days 20d + cost of mileage = 25 = total cost 165 The equation 20d + 25 = 165 represents the situation. 10-3 Solving Two-Step Equations Check It Out: Example 2B Jack rented a car while they were on vacation. He paid a rental fee of $20.00 per day, plus 20¢ a mile. He paid $25.00 for mileage and his total bill for renting the car was $165.00. For how many days did he rent the car? Solve the equation. 20d + 25 = 165 – 25 – 25 20d = 140 20d = 140 20 20 d =7 Subtract 25 from both sides. Divide both sides by 20. Jack rented the car for 7 days. 10-3 Solving Two-Step Equations Additional Example 3: Working Backward with Function Rules The rule for a certain function is to multiply the input by 5 and subtract 4. Find the input value when the output is 11. The function rule is 5 times input minus 4 equals output 5 x x – 4 = y 10-3 Solving Two-Step Equations Additional Example 3 Continued The rule for a certain function is to multiply the input by 5 and subtract 4. Find the input value when the output is 11. 5x – 4 = 11 +4 5x +4 Use the function rule and the given output value to write an equation. Add 4 to both sides to undo the subtraction. = 15 5x = 15 5 5 x =3 The input value is 3. Divide both sides by 5. 10-3 Solving Two-Step Equations Additional Example 3 Continued The rule for a certain function is to multiply the input by 5 and subtract 4. Find the input value when the output is 11. Check Substitute the input value into the rule. 5(3) – 4 = 15 – 4 = 11 check 10-3 Solving Two-Step Equations Check it Out: Example 3 The rule for a certain function is to multiply the input by 6 and subtract 3. Find the input value when the output is 9. The function rule is 6 times input minus 3 equals output 6 x x – 3 = y 10-3 Solving Two-Step Equations Check it Out Example 3 Continued The rule for a certain function is to multiply the input by 6 and subtract 3. Find the input value when the output is 9. 6x – 3 = +3 6x 9 +3 Use the function rule and the given output value to write an equation. Add 3 to both sides to undo the subtraction. = 12 6x = 12 6 6 x =2 The input value is 2. Divide both sides by 6. 10-3 Solving Two-Step Equations Check it Out Example 3 Continued The rule for a certain function is to multiply the input by 6 and subtract 3. Find the input value when the output is 9. Check Substitute the input value into the rule. 6(2) – 3 = 12 – 3 = 9 check 10-3 Solving Two-Step Equations Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 10-3 Solving Two-Step Equations Lesson Quiz Solve each equation. Check your answer. x 1. 12 – 7 = 1 x = 96 2. 22x + 11 = 143 x=6 3. x + 9 = 7 x = 26 4. 9x – 6 = 39 x=5 5 5. In two days, Yasmine drove 505 miles to visit her cousin. The first day, she drove 230 miles. The next day Yasmine drove 5 hours at a constant speed. How fast did Yasmine drive on the second day? 55 mi/h 10-3 Solving Two-Step Equations Lesson Quiz for Student Response Systems 1. Solve 4x + 3 = 27. A. x = 6 B. x = 7.5 B. x = 8 B. x = 96 10-3 Solving Two-Step Equations Lesson Quiz for Student Response Systems 3. Solve A. x = 15 B. x = 20 C. x = 60 D. x = 100 .