Applications of Linear Systems

In class, we worked through this sequence of questions in random groups at the whiteboards.

If you were away

Work through the questions below in order, on paper, before looking at any answers. Each question builds on the one before it. Surprised by what you discover in a given question? That is where the learning happens.

Once you have an answer, expand the answer box to check your work. If a question didn’t go the way you expected, try to explain why before moving on.

When you reach the end, read the summary of key ideas that we built together in class.

TIP

When you solve a system, check your solution by substituting it into both equations. Then answer the question that was asked, in a sentence, with units.

Question 1

Let $x$ represent the larger of two numbers, and $y$ the smaller. Write an expression for:

a) the sum of the two numbers

b) the smaller subtracted from the larger

c) the sum of six times the larger and two times the smaller

Answers

a) $x+y$

b) $x-y$

c) $6x+2y$

“Sum” means add, “subtracted from” means take away (the smaller is taken away from the larger, so the larger comes first), and “times” means multiply.

Question 2

A plane flies at $x$ km/h in still air. The wind speed is $y$ km/h. Write an expression for the speed of the plane when it is flying:

a) into the wind

b) with the wind

A fitness centre charges an initiation fee of $$x$ plus a monthly charge of $$y$. Write an expression for the cost of being a member for:

c) 7 months

d) 15 months

Answers

a) $x-y$

b) $x+y$

c) $x+7y$

d) $x+15y$

The situation decides the operation. A headwind slows the plane down and a tailwind speeds it up. The initiation fee is paid only once, so it is not multiplied; the monthly charge is paid every month, so it is multiplied by the number of months.

Question 3

Tickets to a college swim meet cost $10 for general admission and $5 for students. There were $x$ general admission tickets and $y$ student tickets sold. Write an expression for:

a) the total number of tickets sold

b) the revenue, in dollars, from general admission tickets

c) the revenue, in dollars, from student tickets

d) the total revenue from all the tickets

Answers

a) $x+y$

b) $10x$

c) $5y$

d) $10x+5y$

Notice the difference between how many (a count of tickets) and how much (money). The money from one kind of ticket is its price times the number sold.

Question 4

Traci invested $$x$ at 7% and $$y$ at 6%.

a) Write 7% and 6% as decimals.

Write an expression for:

b) the total amount Traci invested

c) the interest earned at 7% in one year

d) the interest earned at 6% in one year

e) the total interest earned in one year

Answers

a) $0.07$ and $0.06$

b) $x+y$

c) $0.07x$

d) $0.06y$

e) $0.07x+0.06y$

This has the same structure as the tickets in question 3: the interest rate plays the role of the price, and the amount invested plays the role of the number of tickets.

Question 5

Let $x$ represent the larger of two numbers, and $y$ the smaller.

The sum of the two numbers is 20. The smaller subtracted from the larger is 4.

Write two equations. Then find the two numbers.

Answer

$\begin{array}{rl}x+y& =20\\ x-y& =4\end{array}$

Adding the equations gives $2x=24$, so $x=12$, and then $y=20-12=8$.

The numbers are 12 and 8.

The expressions from question 1 become equations once we know what they are equal to.

Question 6

A plane flies at $x$ km/h in still air. The wind speed is $y$ km/h.

Flying into the wind, the plane travels at 450 km/h. Flying with the wind, it travels at 550 km/h.

Write two equations. Then find the speed of the plane in still air and the speed of the wind.

Answer

$\begin{array}{rl}x-y& =450\\ x+y& =550\end{array}$

Adding the equations gives $2x=1000$, so $x=500$, and then $y=550-500=50$.

The plane flies at 500 km/h in still air, and the wind speed is 50 km/h.

Each sentence of the problem gave us one equation.

Question 7

A fitness centre charges an initiation fee of $$x$ plus a monthly charge of $$y$.

Being a member for 7 months costs $255. Being a member for 15 months costs $455.

Write two equations. Then find the initiation fee and the monthly charge.

Answer

$\begin{array}{rl}x+7y& =255\\ x+15y& =455\end{array}$

Subtracting the first equation from the second eliminates $x$: $8y=200$, so $y=25$. Then $x=255-7(25)=80$.

The initiation fee is $80 and the monthly charge is $25.

Check: $80+15(25)=455$.

Question 8

Tickets to a college swim meet cost $10 for general admission and $5 for students. There were $x$ general admission tickets and $y$ student tickets sold.

In total, 350 tickets were sold, and the total revenue was $2750.

Write two equations. Then find how many tickets of each kind were sold.

Answer

$\begin{array}{rl}x+y& =350\\ 10x+5y& =2750\end{array}$

One equation counts tickets; the other counts money. From the first equation, $y=350-x$. Substituting, $10x+5(350-x)=2750$, so $5x+1750=2750$, $x=200$ and $y=150$.

200 general admission tickets and 150 student tickets were sold.

Check: $200+150=350$ and $10(200)+5(150)=2750$.

Question 9

At a banquet hall, two dishes will be served at a wedding: a chicken dish that costs $16 and a beef dish that costs $18. Each of the 300 guests attending is served one dish, and the total cost of preparing the food is $5256. How many chicken dishes and how many beef dishes need to be prepared?

Do not solve yet. What are you asked to find? Write two “let” statements, then two equations.

Answer

We are asked to find the number of chicken dishes and the number of beef dishes. Those are the unknowns.

Let $x$ be the number of chicken dishes.

Let $y$ be the number of beef dishes.

$\begin{array}{rl}x+y& =300\\ 16x+18y& =5256\end{array}$

The first equation counts dishes; the second counts the cost of preparing them.

Question 10

At a wedding, each of 300 guests is served one dish: chicken, which costs $16, or beef, which costs $18. The food costs $5256 in total. With $x$ chicken dishes and $y$ beef dishes:

$\begin{array}{rl}x+y& =300\\ 16x+18y& =5256\end{array}$

Solve the system and check your solution. How many dishes of each kind need to be prepared?

Answer

The solution is $(72,228)$.

From the first equation, $x=300-y$. Substituting, $16(300-y)+18y=5256$, so $4800+2y=5256$, $2y=456$ and $y=228$. Then $x=300-228=72$.

Check: $72+228=300$ and $16(72)+18(228)=1152+4104=5256$.

$\therefore$ 72 chicken dishes and 228 beef dishes need to be prepared.

Question 11

Traci invested $5000 in total. She invested part of it at 7% and the rest at 6%. After one year, she had earned $320 in interest altogether.

How much did Traci invest at each rate?

Write “let” statements and two equations, then solve.

Answer

Let $x$ be the amount, in dollars, invested at 7%.

Let $y$ be the amount, in dollars, invested at 6%.

$\begin{array}{rl}x+y& =5000\\ 0.07x+0.06y& =320\end{array}$

From the first equation, $y=5000-x$. Substituting, $0.07x+0.06(5000-x)=320$, so $0.01x+300=320$ and $x=2000$. Then $y=3000$.

Check: $0.07(2000)+0.06(3000)=140+180=320$.

$\therefore$ Traci invested $2000 at 7% and $3000 at 6%.

The next question was a bonus card, for groups that finished early.

Question 12

Swim meet tickets cost $10 for general admission and $5 for students. In total, 400 tickets were sold for $3100.

With $x$ general admission tickets and $y$ student tickets sold, a group wrote:

$\begin{array}{rl}10x+5y& =400\\ x+y& =3100\end{array}$

What went wrong? Fix it and solve.

Answer

The totals are swapped. $10x+5y$ is an amount of money, so it must equal the revenue, $3100. $x+y$ is a number of tickets, so it must equal 400.

$\begin{array}{rl}x+y& =400\\ 10x+5y& =3100\end{array}$

From the first equation, $y=400-x$. Substituting, $10x+5(400-x)=3100$, so $5x+2000=3100$, $x=220$ and $y=180$.

Check: $220+180=400$ and $10(220)+5(180)=2200+900=3100$.

$\therefore$ 220 general admission tickets and 180 student tickets were sold.

Summary of key ideas

Here is the summary of the concepts we consolidated together in class, after the whiteboard work:

Applications of Linear Systems

NOTE

If the whiteboard conversation didn’t quite click, or you were away, this is the document to read for a concrete summary of the key ideas: how words become mathematical expressions, and the three steps for solving a real-world problem with a linear system (let statements, two equations, solve).

Curriculum connection

B1.2

solve problems that arise from realistic situations described in words or represented by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method (Sample problem: The Robotics Club raised $5000 to build a robot for a future competition. The club invested part of the money in an account that paid 4% annual interest, and the rest in a government bond that paid 3.5% simple interest per year. After one year, the club earned a total of $190 in interest. How much was invested at each rate? Verify your result.).

Link to original