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# Applications of Linear Systems

## Word problems using two equations and two unknowns

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Applications of Systems of Equations

Can you determine two numbers such that the sum of the numbers is 763 and the difference between the same two numbers is 179? How can a system of equations help you?

### Guidance

A system of linear equations can be used to represent real-world problems. A system of equations is used when there are two variables and you are given two pieces of information about how those variables are related. The examples below will show you some common problems that can be solved using a system of equations.

#### Example A

The length of a rectangular plot of land is 255 yards longer than the width. If the perimeter is 1206 yards, find the dimensions of the rectangle.

Solution: The perimeter of a rectangle is found by the formula P=2l+2w\begin{align*}P=2l+2w\end{align*} where P\begin{align*}P\end{align*} is the perimeter, l\begin{align*}l\end{align*} is the length and w\begin{align*}w\end{align*} is the width. The quantities are the length and the width of the rectangle.

• Let the length of the rectangle be represented by ‘l\begin{align*}l\end{align*}’.
• Let the width of the rectangle be represented by ‘w\begin{align*}w\end{align*}’.

The equations would be:

• The length is 255 yards longer than the width
l=w+255
• The perimeter is 1206 yards
2l+2w=1206

The system of equations can now be solved to determine the dimensions of the rectangle. In the first equation the length is expressed in terms of the width. Substitution will be used to solve the system of equations.

{l=w+2552l+2w=1206}2l+2w=12062(w+255)+2w=12062w+510+2w=12064w+510=12064w+510510=12065104w=6964w4=69644w4=6961744w=174l=w+255l=174+255l=429Substitute (w+255) for l in the equation.Apply the distributive property.SimplifySolve the equation.Substitute the value for w into the equation.Solve the equation.

The length of the rectangular plot of land is 429 yards and the width is 174 yards.

#### Example B

Maria had $12100 to invest. She decided to invest her money in bonds and mutual funds. She invested a portion of the money in bonds paying 8% interest per year and the remainder in a mutual fund paying 9% per year. After one year the total income she had earned from the investments was$1043. How much had she invested at each rate?

Solution: The two quantities in this problem are the amount she had invested in bonds and the amount she had invested in mutual funds.

• Let the amount invested in bonds be represented by ‘b\begin{align*}b\end{align*}’.
• Let the amount invested in mutual funds be represented by ‘m\begin{align*}m\end{align*}’.

The equations would be:

• The total amount of money she had to invest was
$12,100b+m=12,100 • The amount of money she earned from the investments was$10430.08b+0.09m=1043

The system of equations will be solved using elimination.

{b+m=12,1000.08b+0.09m=1043}

The first equation will be multiplied by (–0.08)

0.08(b+m=12,100)0.08b0.08m=9680.08b0.08m=968

The equations now have opposite, numerical coefficients for the variable b\begin{align*}b\end{align*}. Add the equations to eliminate the variable b\begin{align*}b\end{align*}.

0.08b0.08m=9680.08b+0.09m=10430.01m=750.01m=750.01m0.01=750.010.01m0.01=7575000.01m=7500b+m=12,100b+7500=12,100b+75007500=12,1007500b=4600Solve the equation.Substitute the value for m into the equation.Solve the equation.

Maria invested $4600 in bonds and$7500 in mutual funds.

#### Example C

Pedro was saving quarters and dimes to buy a new skateboard. After months of saving his coins in a bottle, he emptied its contents and counted the money. The 561 coins in the bottle had a total value of 107.85. How many of each coin were in the bottle? Solution: The two quantities in this problem are the number of quarters and the number of dimes. • Let the number of quarters be represented by ‘q\begin{align*}q\end{align*}’. • Let the number of dimes be represented by ‘d\begin{align*}d\end{align*}’. The equations would be: • The total number of coins in the bottle was 561q+d=561 • The amount of money in the bottle was107.850.25q+0.10d=107.85

This system will be solved using the substitution method. Solve the first equation in terms of quarters.

q+d=561q+dd=561dq=561d0.25q+0.10d=107.850.25(561d)+0.10d=107.85140.250.25d+0.10d=107.85140.250.15d=107.85140.25140.250.15d=107.85140.250.15d=32.400.15d0.15=32.400.150.15d0.15=32.402160.15d=216q+d=561q+216=561q+216216=561216q=345q=345Substitute the value for q into the equation.Apply the distributive property.Solve the equation.Substitute the value for d into the equation.Solve the equation.

The number of quarters that Pedro had saved in the bottle was 345 and the number of dimes was 216.

#### Example D

The Bayplex and Centre 200 rent their ice out to the community whenever possible. The Bayplex charges a flat rate of $20.00 plus$15.00 for every hour rented. Centre 200 charges $50.00 for a flat rate but only asks for$10.00 for every hour rented.

a) Write an equation to model the cost of renting the ice surface for each arena.

b) Determine the intersection point of the equations. What does this intersection point represent?

c) Explain when it is best to use the Bayplex and when it is best to use Center 200.

Solution:

a) Begin by writing the equations to model the cost of renting the ice surface in each arena.

• The cost of renting the ice at the Bayplex
• The cost of renting the ice at Centre

b) The intersection point of the costs for the arenas can be determined by using the comparison method. Both equations are equal to the variable ‘\begin{align*}c\end{align*}’.

Substitute the value for ‘\begin{align*}h\end{align*}’ into one of the original equations.

The intersection point of the system of linear equations is (6,110). This point is the time when the cost of renting the arena will be the same. Renting the Bayplex for six hours will cost $110 and renting Centre 200 for six hours will cost$110.

The amount of interest from the 8% college fund is $60 more than the amount of interest from the 7% retirement fund. The system of equations will be solved by substitution. ### Practice Write each of the following statements as a linear equation in two variables. 1. The sum of two numbers is 100. 2. When six times the larger of two numbers is added to 4 times the smaller, the result is 112. 3. The length of a rectangle is 8m more than 7 times its width. 4. Four times the number of nickels less three times the number of pennies is 56. 5. Jason has some$5 bills and some $1 bills which have a total of$91.

Solve each of the following problems using \begin{align*}2 \times 2\end{align*} systems of linear equations.

1. At a party, there were 72 people. The hostess counted the shoes and found that there were 32 more pairs of ladies’ shoes compared to the number of pairs of males’ shoes. How many males and females were at the party?
2. Two weight loss programs offer competitive services. Super Slim charges $33 to join and$1.50 per session whereas Think Thin charges $2.50 per session and$15 to join. Determine algebraically, under what circumstances you would choose each plan.
3. Today Sam is twice Jenny’s age. Three years ago the sum of their ages was 45. How old are Sam and Jenny today?
4. The parking lot at a local amusement park contained 123 vehicles (cars and buses). Each car is charged $3 to park for the day and each bus is charged$10. If the total revenue for the day was $481.00, how many cars were on the parking lot? 5. Seven times the larger of two numbers less three times the smaller is 351. Six times the larger less twice the smaller is 342. What are the numbers? 6. The debate team washed cars to raise money for a trip. They charged$8 for a large car and $5 for a small car. All together they raised$550 and washed 80 cars. How many of each type of car did they wash?
7. Pencils cost $0.10 each and notebooks cost$2 each. You buy 15 items and spend $9.10. How many pencils did you buy? How many notebooks did you buy? 8. The sum of two numbers is 15 and the product of the same two numbers is 36. What are the numbers? 9. Cereal costs$3.50 a box and milk costs $2.79 a gallon. Suppose you buy five items and spend$16.08. How many boxes of cereal did you buy and how many gallons of milk did you buy?
10. Twice the sum of two number is 72 while the difference between the two numbers is 22. What are the numbers?

### Vocabulary Language: English

elimination

elimination

The elimination method for solving a system of two equations involves combining the two equations in order to produce one equation in one variable.
linear equation

linear equation

A linear equation is an equation between two variables that produces a straight line when graphed.
Linear System of Equation(s)

Linear System of Equation(s)

A linear system of equations is a set of equations that must be solved together to find the one solution that fits them both.
system of equations

system of equations

A system of equations is a set of two or more equations.

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