Mrs. Aitken's Integrated 1
2-8 Solving Equations: Undoing
If you divide a number by 3 and subtract 64, you get 76. What is the number?
- Working backwards
- Start with 76
- Add 64 (undo subtracting 64 by adding 64 )
- 76 + 64 = 140
- Multiply by 3 (undo dividing by 3 by multiplying by 3)
- 140 • 3 = 420 •Check your answer √
This method of solving is called solving by undoing or solving by inverse operations.
- Subtraction is the inverse operation for addition.
- Division is the inverse operation for multiplication.
Examples – Solve the equations
- If you divide a number by 3 and subtract 64, you get 76. What is the number?
- Let x = the number
- (x/3) - 64 = 76 (write the equation)
- (x/3) - 64+ 64 = 76 + 64 (undo subtraction by adding 64 to both sides)
- (x/3) = 140 (simplify equation)
- 3 · (x/3) = 140 · 3 (undo division by multipliny both sides by 3)
- x = 420
- Check your answer
- (420/3) - 64 = 76
- The total cost for 3 lb. of grapes and a 98¢ melon is $2.75. How much do the grapes cost per pound?
- Write and solve an equation.
- Let g = cost per pound for grapes.
- 3g + 0.98 = 2.75 (cost of grapes + cost of melon = total cost)
- 3g + 0.98 – 0.98 = 2.75 – 0.98 (Undo the addition of 0.98 by subtracting 0.98 from both sides.)
- 3g = 1.77
- 3g/3 = 1.77/3 (Undo multiplication by 3 by dividing both sides by 3)
- g = 0.59
- The grapes cost 59¢ per pound.
- Taylor’s lunch cost her $4.80. This amount included the price of the lunch and 5% sales tax and a tip of 15% of the price of the lunch. What was the price of her lunch?
- Write and solve an equation.
- Let p = price of her lunch.
- p + 0.05p + 0.15p = 4.80 (price of lunch + tax + tip = total cost)
- 1.20p = 4.80 (Combine like terms.)
- 1.20p/1.20 = 4.80/1.20 (Undo multiplication by 1.20 by dividing both sides by 1.20)
- p = 4.00
- Taylor’s lunch was $4.00 before tax and tip.
- Rosalie spent $4.72 to buy and mail eight postcards. The total cost of the stamps was $1.52. How much did she pay for each postcard?
- Let p = the cost for a postcard
- 8p + 1.52 = 4.72
- 8p + 1.52 – 1.52 = 4.72 – 1.52
- 8p = 3.2
- p = 0.40
- The postcards cost 40¢ each.
- Check your answer
- 8(40) + 1.52 = 4.72 √
- Zach took 3.5 hours to mow the Smith’s lawn and 1.5 hours to mow the Jone’s lawn. He charged the same amount per hour for each job. In all, he earned $39. What did he charge per hour for this work?
- Let c = cost per hour
- 3.5c + 1.5c = 39
- 5c = 39
- c = 7.80
- Zach charged $7.80 per hour.
- Check your answer
- 3.5(7.8) + 1.5 (7.8) = √
- Some generous person likes to pick apples but doesn’t much enjoy eating them. So she picks a sackful of apples and then proceeds to give most of them away. Each person who receives apples gets a whole number of apples; no apples are cut. She does this in the following way: The first person she meets gets half the apples in the sack. The second person gets half the remaining apples. The third person gets half the remaining apples. The fourth person gets half the remaining apples. The generous person has one apple left. How many apples were in the sack?
- This puzzle can be solved by working backward, undoing the process of giving away the apples.
- Extra credit for solving the apple problem.
Homework
- Read pg. 105-108
- Pg. 108 #8, 12, 15-22
- Practice 16 #1-31