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Core Mathematics

Katherine Yoshiwara

Section 5.4 More Equations

  • Equations with Two Operations
  • A Strategy for Solving Equations
  • Formulas
  • Problem Solving
  • Using Algebra to Solve Problems

Subsection 5.4.1 Equations with Two Operations

In this Lesson we learn how to solve equations with two or more operations.

Activity 5.4.1. Equations with Two Operations.

Emma can order T-shirts from a catalog for $5 each. She pays a $4 handling fee on her order.
  1. Write an algebraic expression for the cost of ordering \(n\) T-shirts.
  2. Write an equation for the number of T-shirts Emma can order with $34.
  3. Describe in words how to find the number of T-shirts Emma can order with $34. Your method should have two steps.
    (Hint: After she pays the handling fee, how much money is left to pay for T-shirts?
In part (3) of Activity 5.4.1 you described a method for solving the equation \(5n + 4 = 34\text{.}\)

Example 5.4.1.

Solve the equation \(~5n + 4 = 34\)

Solution.

Just as we did with one-step equations, we try to isolate the variable on one side of the equation. We will use two steps to isolate the variable. Here is the first step.
\begin{align*} 5n + 4 \amp = 34~~~~~~~~~~~~~~~\blert{\text{First, subtract 4 from both sides.}}\\ \underline{\hphantom{00}\blert{-4}} \amp = \underline{\blert{-4}}\\ 5n \amp = 30 \end{align*}
Here is the second step.
\begin{align*} \dfrac{5n}{\blert{5}} \amp = \dfrac{30}{\blert{5}}~~~~~~~~~~~~~~~\blert{\text{Next, divide both sides by 5.}}\\ n \amp = 6 \end{align*}
To check the solution, we substitute \(n= \alert{6}\) into the original equation.
\begin{align*} \text{Check:} ~~~~ 5(\alert{6}) + 4 \amp =34 ?\\ 30 + 4 \amp = 34 ~~~~\blert{\text{True}} \end{align*}
The solution checks. Emma can order 6 T-shirts with $34.

Checkpoint 5.4.2.

Phoebe is stacking books on top of a footstoolthat is 30 centimeters tall . Each book is 3 centimeters thick.
  1. Write an algebraic expression for the height of the top of a stack of \(n\) books.
  2. Write an equation for the number of books Phoebe needs for the stack to be 54 centimeters high.
  3. Describe how to solve your equation from part (b) in two steps.
Answer.
  1. \(\displaystyle 30 + 3n\)
  2. \(\displaystyle 30 + 3n = 54\)
  3. Subtract 30 from both sides, then divide both sides by 3.

Subsection 5.4.2 A Strategy for Solving Equations

You would like to give a bracelet to a friend as a surprise gift for her birthday. First you wrap the bracelet in a small box, and then you wrap the small box in a large box.
To open the gift, your friend will first unwrap the large box, and then unwrap the small box. That is, she will undo the wrappings in reverse order. We use this same idea to solve equations.
Consider the equation
\begin{equation*} 2x-5=13 \end{equation*}
\(\blert{\text{Question:}}\) What operations have been performed on the variable, \(x\text{?}\)
\(\blert{\text{Answer:}}\) Imagine that you want to evaluate the expression \(2x-5\text{.}\) Following the order of operations, you would:
  1. First multiply \(x\) by 2,
  2. then subtract 5 from the result.
To isolate the variable and solve the equation, we must undo these operations in reverse order. Whatever operation was performed last, we must undo first. So we first add 5 to both sides of the equation, and then divide both sides by 2.
Operations performed on \(x\) Steps for Solution
1. Multiplied by 2 1. Add 5
2. Subtracted 5 2. Divide by 2
Now let’s use our plan to solve the equation.
Step 1
\begin{align*} 2x - 5 \amp = 13 ~~~~~~~~ \blert{\text{Add 5 to both sides of the equation.}}\\ \underline{\hphantom{00}\blert{+5}} \amp = \underline{\blert{+5}}\\ 2x \amp = 18 \end{align*}
Step 2
\begin{align*} \dfrac{2x}{\blert{2}} \amp = \dfrac{18}{\blert{2}} ~~~~~~~~\blert{\text{Divide both sides of the equation by 2.}}\\ x \amp = 9 \end{align*}
To check the solution, we substitute \(\alert{9}\) for \(x\) in the original equation.
\begin{align*} \text{Check:} ~~~~ 2(\alert{9}) - 5 \amp = 13 ?\\ 18 - 5 \amp = 13 ~~~~\blert{\text{True}} \end{align*}
The solution checks.
Here is a summary of our strategy for solving equations.

To solve an equation.

  1. List the operations performed on the variable in order.
  2. Undo those operations in reverse order.

Example 5.4.3.

Solve the equation \(~4 + \dfrac{a}{3} = 9\)

Solution.

Ask yourself what operations were performed on the variable, \(a\text{.}\) Then make a plan to undo those operations in reverse order.
Operations performed on \(x\) Steps for Solution
1. Divided by 3 1. Subtract 4
2. Added 4 2. Multiply by 3
Now we follow the steps in the plan.
Step 1
\begin{align*} 4 + \dfrac{a}{3} \amp = ~~~9 ~~~~~~~~ \blert{\text{Subtract 4 from both sides of the equation.}}\\ \underline{\blert{-4}\hphantom{0000}} \amp = \underline{\blert{-4}}\\ \dfrac{a}{3} \amp = 5 \end{align*}
Step 2
\begin{align*} \blert{3}\left(\dfrac{a}{3}\right) \amp = \blert{3}(5) ~~~~~~~~\blert{\text{Multiply both sides of the equation by 3.}}\\ a \amp = 15 \end{align*}
The solution is 15.
\begin{align*} \text{Check:} ~~~~ 4 +\dfrac{\alert{15}}{3}\amp = 9 ?\\ 4 + 5 \amp = 9 ~~~~\blert{\text{True}} \end{align*}

Checkpoint 5.4.4.

Solve the equation \(~3b-4 = 14\)
Operations performed on \(x\) Steps for Solution
1. 1.
2. 2.
Answer.
6

Note 5.4.5.

With a little practice, you will be able to see the steps to "unwrap" the variable without writing them down. Just remember to undo the last operation first!
In the next example we use three steps to solve the equation.

Example 5.4.6.

Solve the equation \(~\dfrac{5t-3}{6} = 17\)

Solution.

We analyze the order of operations performed on the variable, \(t\text{.}\) Then we make a plan for undoing those operations in the reverse order.
Operations performed on \(t\) Steps for Solution
1. Multiplied by 5 1. Multiply by 6
2. Subtracted 3 2. Add 3
3. Divided by 6 3. Divide by 5
Now we follow the steps in the plan.
\begin{align*} \blert{6} \left(\dfrac{5t-3}{6}\right) \amp = \blert{6}(17) ~~~~~~~~ \blert{\text{Multiply both sides of the equation by 6.}}\\ \\ 5t-3 \amp = 102~~~~~~~~~~~ \blert{\text{Add 3 to both both sides of the equation.}}\\ \underline{\hphantom{00}\blert{+3}} \amp = \underline{\blert{+3}}\\ 5t \amp = 105\\ \\ \dfrac{5t}{\blert{5}} \amp = \dfrac{105}{\blert{5}} ~~~~~~~~ \blert{\text{Divide both sides by 5.}}\\ t \amp = 21 \end{align*}
The solution is 21.
\begin{align*} \text{Check:} ~~~~ \dfrac{5(\alert{21}) - 3}{6}\amp = 17 ?\\ \dfrac{205-3}{6} \amp = 17 ?\\ \dfrac{102}{6} \amp = 17 ~~~~\blert{\text{True}} \end{align*}

Checkpoint 5.4.7.

Solve the equation \(~\dfrac{w + 3}{2} - 5 = 12\)
Operations performed on \(x\) Steps for Solution
1. 1.
2. 2.
3. 3.
Answer.
31

Subsection 5.4.3 Formulas

We use the formula
\begin{equation*} F = \dfrac{9}{5}C + 32 \end{equation*}
to convert temperatures in degrees Celsius to degrees Fahrenheit. By solving an equation, we can also convert from Fahrenheit to Celsius.

Example 5.4.8.

Normal body temperature for people is 98.6\(\degree\) F. Use the formula to convert normal temperature to degrees Celsius.

Solution.

To start, we substitute \(\alert{98.6}\) for \(F\) in the formula.
\begin{equation*} \alert{98.6} = \dfrac{9}{5}C + 32 \end{equation*}
Next, we solve the equation for \(C\text{.}\) We undo the operations in three steps.
Operations performed on \(C\) Steps for Solution
1. Multiplied by 9 1. Subtract 32
2. Divided by 5 2. Multiply by 5
3. Added 32 3. Divide by 0
\begin{align*} 98.6 \amp = \dfrac{9}{5}C + 32 ~~~~~~~~ \blert{\text{Subtract 32 from both sides.}}\\ \underline{\blert{-32}} \amp = \underline{\hphantom{0000}\blert{-32}}\\ 66.6 \amp = \dfrac{9}{5}C ~~~~~~~~~~~~~~~~\blert{\text{Multiply both sides by 5.}}\\ \\ \alert{5}(66.6) \amp = \alert{5}\left(\dfrac{9}{5}C\right) \\ 333 \amp = 9C~~~~~~~~~~~~~~~~~~~ \blert{\text{Divide both sides by 9.}}\\ \\ \dfrac{333}{\blert{9}} \amp = \dfrac{9C}{\blert{9}}\\ 37 \amp = C \end{align*}
Normal body temperature is 37\(\degree\)C. You can check the solution by substituting \(\alert{37}\) for \(C\) in the equation.
\begin{equation*} \text{Does} ~~98.6 = \dfrac{9}{5}(\alert{37}) + 32 ? \end{equation*}

Checkpoint 5.4.9.

The boiling temperature of water is 212\(\degree\)F. Use the formula in the prevopus Example to find the boiling temperature of water in degrees Celsius.
Answer.
\(\degree\)

Subsection 5.4.4 Problem Solving

There are many strategies for solving problems and often the most efficient approach is to write and solve an equation.

Activity 5.4.2. Problem Solving.

Myron is buying a new computer system. He made a down payment of $500 and agreed to make 18 equal monthly payments. If the total price of the system is $6350, how much will each payment be?
In Activity 5.4.2 we consider three ways to solve this problem that do not involve algebra.
\(\blert{\text{Method 1: Guess-and-Check}}\) Guess a value for the amount of each payment, and check to see whether that value gives the correct total price for the computer system. For example, guess that the payment is $200. Myron would then pay
\begin{align*} 500 ~~~~ + ~~~~\amp 18 (200) \amp \amp = ~~??\\ \text{down}~~~~~~~~~~ \amp \text{monthly}~~~~ \amp \amp \text{total}\\ ~~~~\text{payment}~~~~~~~~~~ \amp \text{payments}~~~~ \amp \amp \text{price} \end{align*}
Is the total proice too high or too low? Should you try a larger value or a smaller value for your next guess at the monthly payment? Try your next guess:
\begin{equation*} 500 + 18(\hphantom{000}) = ?? \end{equation*}
We would continue in this way, adjusting our guesses, until we closed in on the correct amount.
\(\blert{\text{Method 2: Make a table.}}\) This is a more organized version of the guess-and-check method. Complete this sample table:
Monthly payment 150 200 250 300 350
Total price \(\hphantom{000000}\) \(\hphantom{000000}\) \(\hphantom{000000}\) \(\hphantom{000000}\) \(\hphantom{000000}\)
Which monthly payments capture the correct total price? We could make another table with smaller increments between those monthly payments. However, this is still a tedious method for solving the problem.
\(\blert{\text{Method 3: Use arithmetic skills.}}\) Since Myron paid $500 down on the system, we can subtract 500 from the total price to find out how much he has left to pay in monthly installments.
\begin{equation*} 6350 - 500 = \alert{??} \end{equation*}
The balance will be paid in 18 equal installments, so we can divide the balance by 18 to find the amount of each installment:
\begin{equation*} \dfrac{\alert{??}}{18} = \end{equation*}
Each monthly payment will be $ .
Method 3 solves the problem efficiently. Notice that it uses the same technique of "undoing" operations that we use to solve equations. If we can describe a problem by an equation, we can work backwards to the solution by solving the equation.

Subsection 5.4.5 Using Algebra to Solve Problems

A Quick Refresher.

\(\blert{\text{Steps for Modeling a Problem}}\)
  1. Identify the unknown quantity and choose a variable to represent it.
  2. Find some quantity that can be expressed in two different ways, and write an equation.
  3. Solve the equation, and answer the question in the problem.

Example 5.4.10.

Here is a solution to the problem in Activity 5.4.2 that uses an equation.

Solution.

Step 1: What are we asked to find in the problem? Choose a variable to represent it.
The amount of each monthly payment: \(p\)
Step 2: We can express the total price of the system in two different ways.
First way: The total price of the system is $6350.
Second way: The total price is the sum of the down payment and all the monthly payments, \(500 + 18p\text{.}\)
Next, we write an equation using our two expressions.
\begin{equation*} 500 + 18p = 6350 \end{equation*}
Now we solve the equation.
\begin{align*} 500 + 18p \amp = 6350 ~~~~~~~~ \blert{\text{Subtract 500 from both sides.}}\\ \underline{\blert{-500}\hphantom{0000}} \amp = \underline{\blert{-500}}\\ 18p \amp = 5850\\ \\ \dfrac{18p}{\blert{18}} \amp = \dfrac{5850}{\blert{18}}~~~~~~~ \blert{\text{Divide both sides by 18.}}\\ 9 \amp = 325 \end{align*}
The solutions is 325. Myron’s monthly payments will be $325.

Note 5.4.11.

Notice that the two steps we used to solve the equation in Example 5.4.10 (subtract 500 and divide by 18) are the same steps we used in Method 3 of Activity 5.4.2, the arithmetic solution.

Checkpoint 5.4.12.

Lloyd scored only 13 out of 20 on his last quiz. That was 5 points lower than his average score over the previous 6 quizzes. How many quiz points did Lloyd accumulate over 6 quizzes?
Step 1: What are we asked to find in the problem? Choose a variable to represent it.
Step 2 Express Lloyd’s score on the last quiz in two different ways, and write an equation using the two expressions.
Step 3: Solve the equation and answer the question in the problem.
Answer.
108 points

Activity 5.4.3. Problem Solving.

Here are two more problems to try.
  1. On the same quiz as Lloyd in the last Exercise 5 Lucy scored 18 out of 20, which was 2 points higher than her average over the previous 6 quizzes. How many points had Lucy accumulated over the first 6 quizzes?
  2. Suppose that Myron’s computer system from the previous Activity goes on sale for $5540 just before he buys it. If he still makes 18 monthly payments after paying $500 down, how much will each of the monthly payments be?

Exercises 5.4.6 Practice 5.4

Exercise Group.

For Problems 1-18, solve the equation. Show the steps in your solution.
1.
\(2x + 5 = 27\)
2.
\(3y + 4 =25\)
3.
\(4a - 6 = 14\)
4.
\(5b-1 = 19\)
5.
\(17 = 7t - 4\)
6.
\(329 = 8w - 3\)
7.
\(21 = 6h + 9\)
8.
\(30 = 9k + 3\)
9.
\(\dfrac{m}{4} - 5 = 2\)
10.
\(\dfrac{n}{7} - 4 = 5\)
11.
\(8 + \dfrac{v}{3} = 10\)
12.
\(2 + \dfrac{u}{6} = 8\)
13.
\(\dfrac{4p}{5} = 8\)
14.
\(\dfrac{7q}{2}= 14\)
15.
\(5(z + 4) = 35\)
16.
\(4(s - 6) = 16\)
17.
\(36 = 9(f-7)\)
18.
\(28 = 7(3 + g)\)

Exercise Group.

Use a calculator as needed to solve the equations in Problems 19-24.
19.
\(5m - 0.35 = 2.4\)
20.
\(6r + 2.8 = 7.6\)
21.
\(0.08d + 5.5 = 13.5\)
22.
\(0.12v - 0.36 = 0.72\)
23.
\(10.2 = \dfrac{w}{1.8} = 2.64\)
24.
\(42.3 = \dfrac{c}{3.5} = 18.7\)

Exercise Group.

For Problems 25-28, choose a variable for each unknown quantity and write algebraic expressions.
25.
The sum of the escrow fees and 1.5% of the selling price
26.
The product of the number of students and three dollars more than last year’s entrance fee
27.
62% of the sum of your test average and your homework average
28.
The sum of your business expenses and 29% of your medical expenses

Exercise Group.

For Problems 29-32,
  1. Complete the table of values.
  2. Use the table to solve the equation.
29.
  1. \(x\) 2 4 6 8 10 12
    \(3x-6\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\)
  2. \(\displaystyle 3x-6 = 12\)
30.
  1. \(x\) 6 8 10 12 14 16
    \(3(x-6)\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\)
  2. \(\displaystyle 3(x-6) = 12\)
31.
  1. \(x\) 4 6 8 10 12 14 16
    \(\dfrac{4+2x}{3}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\)
  2. \(\displaystyle \dfrac{4+2x}{3} = 12\)
32.
  1. \(x\) 4 6 8 10 12 14 16
    \(4+\dfrac{2x}{3}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\)
  2. \(\displaystyle 4+\dfrac{2x}{3} = 12\)

Exercise Group.

For Problems 33-36, choose the correct equation and solve.
\begin{align*} 3n + 12 = 30 \amp \hphantom{0000000} 3n-12 = 30\\ \dfrac{n}{3} + 12 = 30 \amp \hphantom{0000000} \dfrac{n}{3} - 12 = 30 \end{align*}
33.
Zora bought three boxes of Christmas cards and used twelve cards. She has 30 cards left. How many cards were in each box?
34.
Denton read one-third of his history assignment plus 12 pages of economics, for a total of 30 pages. How long is the history assignment?
35.
Cindy planned to bicycle to Sagebrush State Park in three equal daily segments, but a thunder shower came up and she rode only 30 miles the first day, 12 miles shorter than she planned. How far is it to the park?
36.
Margaret makes $3 an hour baby-sitting. If she already has $12, how many hours must she work in order to buy a $30 gift for her mother?

Exercise Group.

For Problems 37-40, choose the correct equation and solve.
\begin{align*} \dfrac{x+20}{4} = 80 \amp \hphantom{0000000} \dfrac{x}{4} +20 = 80\\ 4x+20 = 80 \amp \hphantom{0000000} 4(x-20) = 80 \end{align*}
37.
Delbert bought four college T-shirts and a $20 book for a total of $80. How much did each T-shirt cost?
38.
If Barbara makes just 20 points on the next round of Castles and Kingdoms, her average score over four rounds will be 80 points per round. How many points does she have now?
39.
Francine and her three roommates shared a winning lottery ticket. Adding her share to her $20 savings gives Francine $80. How much did the lottery ticket win?
40.
Four students bought concert tickets with their $20 student discount coupons. They paid a total of $80. How much do the tickets sell for regularly?

Exercise Group.

For Problems 41-48, use the formula to answer the question.
41.
The perimeter of a rectangle is given by \(~\blert{P = 2l + 2w}.~\) Find the length of a rectangle whose perimeter is 86 meters and whose width is 18 meters.
42.
The area of a triangle is given by \(~\blert{A = \frac{1}{2}bh}.~\) Find the height of a triangle whose area is 35 square feet and whose base is 14 feet.
43.
The cost of renting a car for one day is given by \(~\blert{C = d + pm},~\) where \(d\) is the daily fee, \(p\) is the price per mile, and \(m\) is the number of miles driven. One summer day, the daily fee for your rental is $15 and the price per mile is $0.20. If your rental bill was $75 for one day, how far did you drive?
44.
The cost of belonging to a health club is given by \(~\blert{C = I + pm},~\) where \(I\) is the initiation fee, \(p\) is the monthly fee, and \(m\) is the number of months. Amber paid $425 for a one-year membership at Sports Palace. If the initiation fee is $125, how much is the monthly fee?
45.
If you deposit dollars in an account earning simple interest rate \(r\text{,}\) then after \(t\) years the amount of money in the account is given by \(~\blert{A = P + Prt}.~\) Three years ago, Floyd deposited $600 in his account, and he now has $708. What is the interest rate on his account?
46.
If you deposit $5000 in an account that pays 7.5% interest, how long will it be before you have $9500? (Use the formula in Problem 45.)
47.
The area of a trapezoid is given by \(~\blert{A = \dfrac{h}{2}(b + c)},~\) where \(h\) is its height and its bases are \(b\) and \(c\text{.}\) The area of this trapezoid is 54 square inches and its height is 6 inches. If one of the bases is 8 inches long, how long is the other base?
48.
The acceleration of a car is given by \(~\blert{a = \dfrac{f-s}{t}},~\) where \(f\) is the final speed, \(s\) is the starting speed, and \(t\) is the time to reach the final speed. Find \(f\) if \(s=10\text{,}\) \(t=5\text{,}\) and \(a=7\text{.}\)

Exercise Group.

For Problems 49-56,
  1. Write an expression.
  2. Solve an equation.
49.
  1. At 8 am the temperature was 72°, and it has been rising by 6° every hour. At this rate, what will the temperature be after \(h\) hours?
  2. When was the temperature 96?
50.
  1. Avram has typed 480 words of his term paper, and is still typing at a rate of 30 words per minute. How many words will Avram have typed after \(m\) minutes?
  2. When will Avram have typed 1230 words?
51.
  1. For her mother’s retirement party, Daniella spent $50 on a gift plus her share of the cost of the party. If the party costs \(P\) dollars and 12 people are contributing (including Daniella), how much did Daniella spend?
  2. If Daniella spent $68, how much did the party cost?
52.
  1. Delbert shares a house with four roommates. He pays $200 rent per month, plus an equal share of the utilities. If the utilities cost \(U\) dollars this month, how much does Delbert owe?
  2. Delbert paid $263 last month. What was the utility bill for the house?
53.
  1. Brian reserved $60 from the office sunshine fund for a party, and used the rest to buy new desk sets for the 15 people in the office. If the sunshine fund originally had \(S\) dollars, how much can Brian spend on each desk set?
  2. If each desk set cost $23, how much was originally in the sunshine fund?
54.
  1. Vincent’s school provided him with \(C\) colored pencils to distribute to the students in his class, and he bought 8 more pencils so that each student would get the same number. If there are 23 students in Vincent’s class, how many pencils does each get?
  2. If each students got 4 colored pencils, how many pencils did the school provide?
55.
  1. Raylyn’s auto registration fee is $20 plus 2% of the value of her car. If Raylyn’s car is worth \(B\) dollars, how much will her registration fee be?
  2. If Raylyn paid $340 for registration, how much is her car worth?
56.
  1. Maryam has to carry a dummy weighing 50 pounds plus 60% of her own weight for 20 yards as part of the test to become a lifeguard. If Maryam weighs \(W\) pounds, how much must she carry?
  2. If Maryam must carry 134 pounds, how much does she weigh?

Exercise Group.

For Problems, 57-64,
  1. identify the unknown quantity and choose a variable to represent it,
  2. use the hint to help you write an equation,
  3. solve the equation, then write your answer in a sentence.
57.
Irwin bought some DVDs through the mail at a cost of $7 each. He also paid $2 for shipping and handling. If the total cost of Irwin’s order was $30, how many DVDs did he buy? Hint: Write an expression in terms of your variable for the total cost of Irwin’s order.
58.
Cassandra ordered some scrap books as graduation gifts. Each book cost $12, and the total shipping cost was $4. If Cassandra’s order totaled $76, how many scrapbooks did she order? Hint: Write an expression in terms of your variable for the total cost of Cassandra’s order.
59.
Fran wants to buy an $86 espresso maker. She already has $30. If she earns $8 per hour as a cashier, how many hours must she work to earn the money she needs? Hint: Write an expression in terms of your variable for the amount of money Fran has saved.
60.
Darla is saving to buy a pair of roller blades for $111. She already has $27. If she saves $12 per week, how long will it take to save the money she needs? Hint: Write an expression in terms of your variable for the amount of money Darla has saved.
61.
A pair of birding binoculars costs $269. That is $9 more than 4 times the cost of a camera tripod. How much does the tripod cost? Hint: Write an expression in terms of your variable for the cost of the binoculars.
62.
If you divide the cost of a TV by 5 and subtract $13, you get the cost of a pair of headphones. The headphones cost $39. How much does the TV cost? Hint: Write an expression in terms of your variable for the cost of the headphones.
63.
This year Georgia paid $2840 in state taxes. That was $600 plus 7% of her adjusted income. What was Georgia’s adjusted income this year? Hint: Write an expression in terms of your variable for Georgia’s taxes.
64.
Gilbert’s speech instructor will add 5% of his score in the State Debating Contest to his homework total. If Gilbert would like to raise his total from 652 points to 675 points, what must he score in the debate? Hint: Write an expression in terms of your variable for Gilbert’s homework score.

Exercise Group.

For Problems 65-70, simplify each side of the equation, then solve.
65.
\(298 = \dfrac{1}{2}(18)h + 46\)
66.
\(316 = \dfrac{1}{2}(16)g + 4(15)\)
67.
\(1080 = 600 + 600(0.04)t\)
68.
\(1187.50 = 950 + 950(0.05)t\)
69.
\(5w - 3(6.4) = 120 + 0.2(38)\)
70.
\(12 + 50r = 2(173) + 4(85)\)

Exercise Group.

For Problems 71-78, solve the equation by using three steps.
71.
\(\dfrac{2g}{7} + 12 = 16\)
72.
\(\dfrac{5d}{3} - 5 = 10)\)
73.
\(11 = \dfrac{9z}{4} - 7\)
74.
\(21 = \dfrac{3c}{8} + 15\)
75.
\(\dfrac{3y-4}{5} = 4\)
76.
\(\dfrac{2x+5}{7} = 3\)
77.
\(2(5b+1) = 32\)
78.
\(3 (4r-9) = 45\)
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