Analytic Geometry · Analytic Geometry

Modelling Problems with Linear Systems

1.2About 70 minutes

Learning goals

I can define variables and write a linear system for a problem described in words.
I can choose an efficient method to solve the system.
I can check my answer in the words of the problem and write a clear conclusion.
I can interpret a point of intersection to compare two options.

Words to know

let-statement
A sentence that says what each variable stands for, with units.

model
A mathematical description, such as a system of equations, that represents a real situation.

commission
Pay calculated as a percent of the value of sales.

break-even point
The point where two options cost or pay the same; the point of intersection.

reasonable
Makes sense in the real situation (for example, whole numbers of tickets).

From a story to a system

Most real problems do not arrive as two neat equations. They arrive as a story: a school play, a job offer, a fence to build. Your job is to turn the story into a linear system, solve it, and then turn the answer back into words.

Whenever a problem has two unknown quantities and gives you two separate facts that connect them, a linear system is a good model.

A five-step plan

Define the variables with a let-statement, including units. "Let a be the number of adult tickets."
Write two equations, one for each fact. Often one equation counts how many and the other counts how much (money, mass, distance).
Choose a method (graphing, substitution or elimination) and solve.
Check the answer in the words of the problem, not just in your equations. (If your equation is wrong, checking in it will not catch the mistake.)
Conclude with a sentence that answers the question asked.

Translating key phrases into algebra. Each fact in the story becomes one equation (or part of one).

Worked example

Example 1: Tickets for the school play

A school sold 230 tickets to its winter play and collected $2160. Adult tickets cost $12 and student tickets cost $8. How many of each were sold?

Define: let a be the number of adult tickets and s the number of student tickets.
Equations: count of tickets: a + s = 230. Money: 12a + 8s = 2160.
Solve by substitution: s = 230 − a. Then 12a + 8(230 − a) = 2160, so 12a + 1840 − 8a = 2160, 4a = 320, a = 80. Then s = 230 − 80 = 150.
Check in the words: 80 + 150 = 230 tickets ✓. Money: 80 × $12 = $960 and 150 × $8 = $1200; $960 + $1200 = $2160 ✓.

The school sold 80 adult tickets and 150 student tickets.

Worked example

Example 2: Two salary offers

Maria has been hired by a car company to sell cars. Option 1 pays $2000 per month plus 2% commission on her sales. Option 2 pays $1500 per month plus 3% commission. For what monthly sales do the options pay the same? Which option should she choose?

Define: let s be Maria's monthly sales in dollars and E her monthly earnings in dollars.
Equations: Option 1: E = 2000 + 0.02s. Option 2: E = 1500 + 0.03s. (2% = 0.02 and 3% = 0.03.)
Solve by substitution (both are solved for E): 2000 + 0.02s = 1500 + 0.03s. Subtract 1500 and 0.02s from both sides: 500 = 0.01s, so s = 50 000.
Then E = 2000 + 0.02(50 000) = 2000 + 1000 = 3000. Check: 1500 + 0.03(50 000) = 1500 + 1500 = 3000 ✓.
Interpret: test sales of $40 000: Option 1 pays $2800, Option 2 pays $2700. Test $60 000: Option 1 pays $3200, Option 2 pays $3300.

The options pay the same, $3000, when monthly sales are $50 000. If Maria expects to sell more than $50 000 a month, Option 2 pays more; if less, Option 1 pays more.

Maria's two offers. The dashed Option 2 line is steeper (3% commission) but starts lower. The lines cross at sales of $50 000 and earnings of $3000; to the right of that point Option 2 is higher.

Worked example

Example 3: A rectangular garden

The perimeter of a rectangular vegetable garden is 54 m. The length is 3 m more than twice the width. Find the dimensions.

Define: let l be the length and w the width, both in metres.
Equations: 2l + 2w = 54 and l = 2w + 3.
Solve by substitution: 2(2w + 3) + 2w = 54, so 4w + 6 + 2w = 54, 6w = 48, w = 8. Then l = 2(8) + 3 = 19.
Check in the words: 19 is 3 more than twice 8 (16 + 3 = 19) ✓. Perimeter: 2(19) + 2(8) = 38 + 16 = 54 ✓.

The garden is 19 m long and 8 m wide.

Worked example

Example 4: Canoe rentals

Paddle Co. rents canoes for $25 plus $10 per hour. Lakeside Rentals charges $40 plus $7 per hour. When do they cost the same? Which is cheaper for a 3-hour trip? A full 8-hour day?

Define: let h be the number of hours and C the cost in dollars.
Equations: C = 25 + 10h and C = 40 + 7h.
Solve: 25 + 10h = 40 + 7h, so 3h = 15, h = 5. C = 25 + 50 = 75. Check: 40 + 35 = 75 ✓.
3 hours: Paddle Co. $55, Lakeside $61. 8 hours: Paddle Co. $105, Lakeside $96.

They cost the same, $75, for 5 hours. Paddle Co. is cheaper for the 3-hour trip; Lakeside is cheaper for the 8-hour day.

Is the answer reasonable?

Before you write your conclusion, ask: does the answer make sense? Numbers of tickets or people must be whole numbers and not negative. Lengths must be positive. If you get 80.5 tickets, go back and look for an error. Graphs help here too: a quick sketch shows whether the crossing point is in a sensible place.

Just for fun

Word problems: the only place where someone buys 60 watermelons and nobody asks why.

Hands-on activity

Compare two real plans

You need: a grown-up, a flyer, website or bill showing two price plans for the same thing (two phone plans, two gym or pool passes, two car-share or bike-share plans, two delivery options), paper, a pencil and graph paper or Desmos.

With the grown-up, pick two plans that each have a fixed cost and a rate (per month, per visit, per GB, per km...).
Define your variables and write an equation for each plan.
Solve the system with the method of your choice to find where the plans cost the same.
Graph both plans to see which is cheaper on each side of the crossing point.
Estimate how much your family actually uses (visits, GB, km) and write a recommendation in two or three sentences.

What to look for: clear let-statements with units, a correct crossing point that checks in both equations, and a recommendation that uses the family's real usage, not just the crossing point.

Your turn

A movie theatre sold 300 tickets. Adult tickets cost $14 and child tickets cost $9. The total was $3570. How many of each were sold?

The sum of two numbers is 38 and their difference is 12. Find the numbers.

A rectangle's perimeter is 70 cm. Its length is 5 cm more than its width. Find its dimensions.

Jamal has 22 coins, all dimes and quarters, worth $3.70. How many of each does he have?

Salesperson pay: Plan A is $400 per week plus 5% of sales. Plan B is $250 per week plus 8% of sales. For what weekly sales are the plans equal?

A school bought 3 basketballs and 4 volleyballs for $150, then 2 basketballs and 5 volleyballs for $135. Find the price of each ball.

More practice: warm-up

Write a let-statement for: "the number of loonies and toonies in a jar."

Write an equation: there are 40 animals, cows and chickens.

Write an equation: the cows and chickens have 112 legs in total.

Solve the cows-and-chickens system.

Write 4.5% as a decimal.

More practice: on your own

A ski hill sells day passes for $65 and half-day passes for $45. On a snowy Saturday it sold 520 passes for $30 800. How many of each?

Two plans for a summer job: Plan 1 pays $15 per hour. Plan 2 pays $60 per week plus $12 per hour. For how many hours per week are they equal? Which is better for 25 hours?

An isosceles triangle has a perimeter of 50 cm. Each of the two equal sides is 5 cm longer than the base. Find the side lengths.

The difference of two numbers is 7. Three times the smaller plus twice the larger is 59. Find the numbers.

A theatre company charges $25 for adults and $15 for students. A group of 18 paid $370. A student says "9 adults and 9 students". Check this and find the correct answer.

Challenge

A two-digit number has digits that add to 11. If the digits are reversed, the new number is 27 more than the original. Find the number.

A cell plan costs $35 per month and includes 10 GB; extra data costs $5 per GB. Another plan costs $55 per month with unlimited data. For how many GB per month are the costs equal? Explain why this situation is only partly linear.

A farmer's market sells maple syrup in small bottles for $8 and large bottles for $20. One Saturday it sold 4 times as many small bottles as large bottles and took in $1040. How many of each were sold? Show that guess-and-check would be slow but a system is quick.

Check your understanding

Quiz

A store sold 50 shirts, some at $20 and some at $30, for $1200. Which system models this?

In the shirt problem above, how many $30 shirts were sold?

Option 1 pays $2000 + 2% of sales; Option 2 pays $1500 + 3% of sales. At which monthly sales do they pay the same?

Why should you check a word-problem answer in the words of the problem, not only in your equations?

A rectangle has perimeter 40 m and its length is 4 m more than its width. What is its width?

Solving a ticket problem gives 62.5 adult tickets. What should you do?

For parents and teachers

Big idea: turn a story into two equations, solve, check in the story, and answer in a sentence. This lesson teaches expectation 1.2 (solving problems from realistic situations described in words). Example 2 is the salary sample problem from the Ontario curriculum.

How to help: Read word problems aloud together and ask, "What are the two unknowns? What two facts connect them?" Insist on let-statements with units and a final sentence. The real-plan activity is a valuable life skill as well as good math.

Common difficulties:

Mixing up the "how many" and "how much" equations (e.g. writing a + s = 2160).
Using percents as whole numbers (2 instead of 0.02).
Not checking whether the answer is reasonable (a negative number of tickets, or a fraction of a person).
Stopping at the numbers without answering the question in words.