Contemporary Mathematics

# 5.2Linear Equations in One Variable with Applications

Contemporary Mathematics5.2 Linear Equations in One Variable with Applications

Figure 5.4 Most gyms have a monthly membership fee. (credit: modification of work "Morning PT after the Holidays 2021" by Fort Drum & 10th Mountain Division (LI)/Flickr, Public Domain Mark 1.0)

## Learning Objectives

After completing this section, you should be able to:

1. Solve linear equations in one variable using properties of equations.
2. Construct a linear equation to solve applications.
3. Determine equations with no solution or infinitely many solutions.
4. Solve a formula for a given variable.

## Example 5.14

### Constructing a Linear Equation to Solve an Application

The Beaudrie family has two cats, Basil and Max. Together, they weigh 23 pounds. Basil weighs 16 pounds. How much does Max weigh?

1.
Sam and Henry are roommates. Together, they have 68 books. Sam has 26 books. How many books does Henry have?

## Example 5.16

### Constructing an Application from a Linear Equation

Write an application that can be solved using the equation $50x+35=18550x+35=185$. Then solve your application.

1.
Write an application that can be solved using the equation $25x + 75 = 200$. Then solve your application.

## Linear Equations with No Solutions or Infinitely Many Solutions

Every linear equation we have solved thus far has given us one numerical solution. Now we'll look at linear equations for which there are no solutions or infinitely many solutions.

## Example 5.17

### Solving a Linear Equation with No Solution

Solve $3(x+4)=4x+8−x3(x+4)=4x+8−x$.

1.
Solve $2\left( {x + 6} \right) = 3x + 4-\left( {x + 5} \right)$.

## Example 5.18

### Solving a Linear Equation with Infinitely Many Solutions

Solve $2(x+5)=4(x+3)−2x−22(x+5)=4(x+3)−2x−2$.

1.
Solve $3x-7-\left( {x + 5} \right) = 2\left( {x-6} \right).$

## Solving a Formula for a Given Variable

You are probably familiar with some geometry formulas. A formula is a mathematical description of the relationship between variables. Formulas are also used in the sciences, such as chemistry, physics, and biology. In medicine they are used for calculations for dispensing medicine or determining body mass index. Spreadsheet programs rely on formulas to make calculations. It is important to be able to manipulate formulas and solve for specific variables.

To solve a formula for a specific variable means to isolate that variable on one side of the equal sign with a coefficient of 1. All other variables and constants are on the other side of the equal sign. To see how to solve a formula for a specific variable, we will start with the distance, rate, and time formula.

## Example 5.19

### Solving for a Given Variable with Distance, Rate, and Time

Solve the formula $d=rtd=rt$ for $tt$. This is the distance formula where $dd$ = distance, $rr$ = rate, and $tt$ = time.

1.
Solve the formula $I = Prt$ for $t$. This formula is used to calculate simple interest $I$, for a principal $P$, invested at a rate $r$, for $t$ years.

## Example 5.20

### Solving for a Given Variable in the Area Formula for a Triangle

Solve the formula $A=½A=½$ $bhbh$ for $hh$. This is the area formula of a triangle where $AA$ = area, $bb$ = base, and $hh$ = height.

1.
Solve the formula $V = \frac{1}{3}\pi {r^2} h$ for $h$. This formula is used to calculate the volume $V$ of a right circular cone with radius $r$ and height $h$.

## WORK IT OUT

### Using Algebra to Understand Card Tricks

You will need to perform this card trick with another person. Before you begin, the two people must first decide which of the two will be the Dealer and which will be the Partner, as each will do something different. Once you have decided upon that, follow the steps here:

Step 1: Dealer and Partner: Take a regular deck of 52 cards, and remove the face cards and the 10s.

Step 2: Dealer and Partner: Shuffle the remaining cards

Step 3: Dealer and Partner: Select one card each, but keep them face down and don’t look at them yet.

Step 4: Dealer: Look at your card (just the Dealer!). Multiply its value by 2 (Aces = 1).

Step 5: Dealer: Add 2 to this result.

Step 7: Partner: Look at your card.

Step 8: Partner: Calculate: 10 - your card, and tell this information to the dealer.

Step 9: Dealer: Subtract the value the Partner tells you from your total to get a final answer.

Step 10: Dealer: verbally state the final answer.

Step 11: Dealer and Partner: Turn over your cards. Now, answer the following questions

1. Did the trick work? How do you know?
2. Why did this occur? In other words, how does this trick work?

9.
Is the solution strategy used in solving the linear equation correct? If it is correct, show the final step (check the solution). If it is not correct, explain why.
$\begin{array}{rcl}{8\left( {x-2} \right)}&{ = }&{6\left( {x + 10} \right)}\\{8x-16}&{ = }&{6x + 60}\\{8x-16{\mathbf\,{-\,6x}}}&{ = }&{6x + 60{\mathbf\,{-\,6x}}}\\{2x-16 + {\mathbf{16}}}&{ = }&{60 + {\mathbf{16}}}\\{2x}&{ = }&{76}\\{x}&{ = }&{38}\end{array}$
10.
Is the solution strategy used in solving the linear equation correct? If it is correct, show the final step (check the solution). If it is not correct, explain why.
$\begin{array}{rcl}{7 + 4\left( {2 + 5x} \right)}&{ = }&{3\left( {6x + 7} \right)-\left( {13x + 36} \right)}\\{7 + 8 + 20x}&{ = }&{18x + 21-13x-36}\\{15 + 20x}&{ = }&{5x-15}\\{15 + 20x\,{\mathbf{-\,5\mathit{x}}}}&{ = }&{5x-15\,{\mathbf{-\,5\mathit{x}}}}\\{15 + 15x\,{\mathbf{-\,15}}}&{ = }&{- 15\,{\mathbf{-\,15}}}\\{15x}&{ = }&{- \,30}\\{x}&{ = }&{- \,2}\end{array}$
11.
Is the solution strategy used in solving the linear equation correct? If it is correct, show the final step (check the solution). If it is not correct, explain why.
$\begin{array}{rcl}{8x + 7-\left( {2x-9} \right)}&{ = }&{22-\left( {4x-4} \right)}\\{8x + 7-2x-9}&{ = }&{22-4x-4}\\{6x-2}&{ = }&{18-4x}\\{6x-2 +{\mathbf{ 4\mathit{x}}}}&{ = }&{18-4x +{\mathbf{ 4\mathit{x}}}}\\{10x-2 +{\mathbf{ 2}}}&{ = }&{18 +{\mathbf{ 2}}}\\{10x}&{ = }&{20}\\{x}&{ = }&{2}\end{array}$
For the following exercises, use this scenario: The Nice Cab Company charges a flat rate of $3.00 for each fare, plus$1.70 per mile. A competing taxi service, the Enjoyable Cab Company, charges a flat rate of $5.00 for each fare, plus$1.60 per mile.
12.
Using the variable $x$ for number of miles, write the equation that would allow you to find the total fare $(T)$ using the Nice Cab Company.
13.
It is 22 miles from the airport to your hotel. What would be your total fare using the Nice Cab Company?
14.
Using the variable $y$ for number of miles, write the equation that would allow you to find the total fare $(T)$ using the Enjoyable Cab Company.
15.
Using the same 22-mile trip from the airport to the hotel, how much would the total fare be for using the Enjoyable Cab Company?
16.
Based on the cost of each cab ride, which cab company should you use for the trip from the airport to the hotel? Why?
17.
After solving the linear equation $3\left( {2x-3} \right) = 12\left( {x-3} \right)-3\left( {2x-9} \right)$, Nancy says there is no solution. Luis believes there are infinitely many solutions. Who is right?
18.
The conversion formula between the Fahrenheit temperature scale and the Celsius temperature scale is given by this formula: $C = \frac{5}{9}\left( {F-32} \right)$, where $C$ is the temperature in degrees Celsius and $F$ is the temperature in degrees Fahrenheit. What is the correct formula when solved for $F$?
1. $F = \frac{5}{9}C-32$
2. $F = \frac{9}{5}C-32$
3. $F = \frac{5}{9}C + 32$
4. $F = \frac{9}{5}C + 32$
19.
To find a temperature on the Kelvin temperature scale, add 273 degrees to the temperature in Celsius. Which formula illustrates this?
1. $C = K + 273$
2. $K = C + 273$
3. $K = C-273$
4. $C = K-273$
20.
Using the information from exercise 18 and exercise 19, which conversion formula would you use to find degrees Kelvin when given degrees Fahrenheit?
1. $K = \frac{5}{9}\left( {F-32} \right) + 273$
2. $K = \frac{5}{9}F + 241$
3. $K = \frac{9}{5}\left( {F-32} \right) + 273$
4. $K = \frac{9}{5}F + 241$
21.
There is a fourth temperature scale, although it is not used much today. The Rankin temperature scale varies from the Fahrenheit scale by about 460 degrees. So given a temperature in Fahrenheit, add 460 degrees to get the temperature in Rankin. Which formula represents a formula to find degrees Rankin when given degrees Celsius?
1. $R = \frac{5}{9}C-492$
2. $R = \frac{9}{5}C + 492$
3. $R = C + 492$
4. $R = \frac{5}{9}\left( {C-492} \right)$

## Section 5.2 Exercises

For the following exercises, solve the linear equations using a general strategy.
1 .
$- (t - 19) = 28$
2 .
$51 + 5(4 - q) = 56$
3 .
$- 6 + 6(5 - k) = 15$
4 .
$3(10 - 2x) + 54 = 0$
5 .
$- 2(11 - 7x) + 54 = 4$
For the following exercises, solve the linear equations using properties of equations.
6 .
$- 12 + 8(x - 5) = - 4 + 3(5x - 2)$
7 .
$4(p - 4) - (p + 7) = 5(p - 3)$
8 .
$3(a - 2) - (a + 6) = 4(a - 1)$
9 .
$4[5 - 8(4c - 3)] = 12(1 - 13c) - 8$
10 .
$5[9 - 2(6d - 1)] = 11(4 - 10d) - 139$
For the following exercises, construct a linear equation to solve an application.
11 .
It costs $0.55 to mail one first class letter. Construct a linear equation and solve to find how much it costs to mail 13 letters. 12 . Normal yearly snowfall at the local ski resort is 12 inches more than twice the amount it received last season. The normal yearly snowfall is 62 inches. Construct a linear equation and solve to find what the snowfall was last season. 13 . Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was three more than twice the number of notebooks. He bought seven textbooks. Construct a linear equation and solve to find how many notebooks he bought. 14 . Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. Construct a linear equation and solve to find how many crossword puzzles he did. 15 . Laurie has$46,000 invested in stocks. The amount invested in stocks is \$8,000 less than three times the amount invested in bonds. Construct a linear equation and solve to find how much Laurie invested in bonds.
For the following exercises, construct an application from a linear equation.
16 .
$1,000x + 2,500 = 16,500$.
17 .
0.36$t$ for $t = 333$.
18 .
$150\,n + 120 = 570$.
19 .
$4c + 2c$ for $c = 5$.
20 .
$2s + 10 = 24$.
For the following exercises, state whether each equation has exactly one solution, no solution, or infinitely many solutions.
21 .
$23z + 19 = 3(5z - 9) + 8z + 46$
22 .
$18\left( {5j-11} \right) = 47j + 17$
23 .
$22\left( {3\,m + 4} \right) = 17\left( {4\,m-6} \right)$
24 .
$7v + 42 = 11(3v + 8) - 2(13v - 1)$
25 .
$45(3y - 2) = 9(15y - 6)$
26 .
$9\left( {14\,d + 9} \right) + 4\,d = 13\left( {10\,d + 6} \right) + 3$
For the following exercises, solve the given formula for the specified variable.
27 .
Solve the formula $C = \pi d$ for $d$.
28 .
Solve the formula $V = LWH$ for $L$.
29 .
Solve the formula $A = \left( {1/2} \right)bh$ for $b$.
30 .
Solve the formula $A = \left( {1/2} \right) {d_1}{d_2}$ for ${d_1}$.
31 .
Solve the formula $A = \left( {1/2} \right)h\left( {{b_1} + {b_2}} \right)$ for ${b_1}$.
32 .
Solve the formula $h = 54\,t + \left(1/2 \right) a$ for $a$.
33 .
Solve $180{\text{ }} = a + b + c$ for $a$.
34 .
Solve the formula $A = {\text{ }}\left( 1/2 \right)pl + B$ for $p$.
35 .
Solve the formula: $P = 2\,L + 2\,W$ for $L$.