C. Algebra · Algebra: Expressions, Equations and Coding

Creating Equations to Model and Solve Problems

C1.5 C1.2 A1 A2About 75 minutes

Learning goals

I can define a variable clearly, with units, for a real situation.
I can create an equation that models a situation in words, a diagram or a table.
I can solve the equation and verify the answer in the original situation.
I can use equations to compare options and make decisions.

Words to know

let statement
A sentence that defines the variable, such as “Let n be the number of snowfalls.”

model
An equation (or expression, table or graph) that represents a real situation.

break-even point
The value where two options cost the same.

consecutive integers
Integers that follow each other in order, such as 28, 29, 30. They can be written n, n + 1, n + 2.

verify in context
Check that the answer makes sense in the original words of the problem, not just in your equation.

Let's start with a snowy decision

A family is choosing a snow-removal company for the winter.

Company A charges a $150 seasonal fee plus $25 per snowfall.
Company B charges $50 per snowfall, with no seasonal fee.

Which company is the better deal? It depends on how much it snows! An equation will tell us exactly where the two companies cost the same.

A plan for every word problem

Word problems feel hard because there's no equation yet. Creating the equation is the main job. Use these six steps every time.

Six steps for problem solving with equations.

Understand. Read twice. What is asked? What information is given? Draw a sketch if it helps.
Define the variable with a let statement, including units: “Let n be the number of snowfalls.” If there are two unknowns, write the second in terms of the first.
Create the equation. Turn each fact into an expression, then set two equal things equal.
Solve it.
Verify in context. Check the answer against the original words, not just your equation. Does it make sense? (A negative number of tickets doesn't!)
Answer with a sentence that responds to the question.

Worked example

Example 1: The snow-removal decision

When do Company A and Company B cost the same? Which should the family choose?

Let n be the number of snowfalls in the season.
Cost of A: 150 + 25n. Cost of B: 50n.
They cost the same when 150 + 25n = 50n.
Subtract 25n: 150 = 25n. Divide by 25: n = 6.
Verify: A costs 150 + 25(6) = $300. B costs 50(6) = $300 ✓.
With fewer than 6 snowfalls, B is cheaper (for example, 4 snowfalls: A = $250, B = $200). With more than 6, A is cheaper (10 snowfalls: A = $400, B = $500).

They cost the same, $300, at 6 snowfalls. In a snowy part of Ontario where more than 6 snowfalls are likely, Company A is the better deal.

Graphing both cost expressions shows the same answer: the lines cross at (6, 300). To the right of the crossing, Company A's line is lower, so A is cheaper.

Worked example

Example 2: A perimeter problem

A rectangular skating area has a perimeter of 54 m. Its length is 3 m more than twice its width. Find its dimensions.

Let w be the width in metres. Then the length is 2w + 3.
Perimeter: 2w + 2(2w + 3) = 54.
Expand and collect: 2w + 4w + 6 = 54, so 6w + 6 = 54.
6w = 48, so w = 8. Length = 2(8) + 3 = 19.
Verify in context: is 19 three more than twice 8? 2(8) + 3 = 19 ✓. Perimeter: 2(19) + 2(8) = 38 + 16 = 54 ✓.

The skating area is 8 m wide and 19 m long.

Worked example

Example 3: Tickets

A school sold 150 concert tickets. Student tickets cost $8 and adult tickets cost $12. Ticket sales were $1520. How many of each were sold?

Let s be the number of student tickets. Then the number of adult tickets is 150 − s.
Money: 8s + 12(150 − s) = 1520.
Expand: 8s + 1800 − 12s = 1520, so −4s + 1800 = 1520.
−4s = −280, so s = 70. Adults: 150 − 70 = 80.
Verify: 70 + 80 = 150 tickets ✓. 8(70) + 12(80) = 560 + 960 = $1520 ✓.

70 student tickets and 80 adult tickets were sold.

Worked example

Example 4: Angles in a triangle

The angles of a triangle measure x, 2x and x + 20 degrees. Find each angle.

The angles of a triangle add to 180°.
x + 2x + (x + 20) = 180, so 4x + 20 = 180.
4x = 160, so x = 40.
Angles: 40°, 2(40) = 80°, and 40 + 20 = 60°. Verify: 40 + 80 + 60 = 180 ✓.

The angles are 40°, 80° and 60°.

The triangle from Example 4, drawn with its actual angles: 40° (x), 80° (2x) and 60° (x + 20).

Algebra at work

People create equations like these at work all the time. An event planner compares a venue with a flat fee to one that charges per guest. A contractor finds the length of a deck from its perimeter. A small-business owner finds how many items to sell to cover costs. In each case the steps are the same: define, model, solve, check, decide.

Hands-on activity

Find the break-even point at home

You need: paper, a pencil, a calculator, and a real choice your family makes (or use the sample below).

Pick a choice between two options where one has an up-front cost and the other costs more each time. Examples: a monthly transit pass versus paying per ride; buying skates versus renting them each time; a gym membership versus drop-in visits.
Find or estimate the prices. Write an expression for each option's cost after n uses.
Set them equal and solve to find the break-even point.
Verify by calculating both costs at that value.
Write a recommendation: “If we use it more than ___ times, choose ___.”

Sample: a pass costs $100 a month, or $3.50 per ride. 3.50r = 100 gives r ≈ 28.6. Check: 28 rides cost $98 and 29 rides cost $101.50. So with 29 or more rides in a month, the pass is cheaper.

Just for fun

Writing a let statement is like labelling leftovers before they go in the fridge.

Skip it, and three days later nobody remembers what x was.

Your turn

Gym A charges $40 to join plus $15 a month. Gym B charges $25 a month and nothing to join. After how many months do they cost the same?

A rectangle's length is 5 cm more than its width, and its perimeter is 58 cm. Find its dimensions.

The sum of three consecutive integers is 102. Find them.

The angles of a triangle are x, x + 10 and 2x degrees. Find each angle.

A play sold 200 tickets: students paid $5 and adults $9. Sales were $1360. How many of each?

A club has $120 and earns $35 per hour at a car wash. How many hours must it run to reach $400?

Lena is 3 times as old as her brother. In 4 years, the sum of their ages will be 40. How old is each now?

More practice: warm-up

Write and solve: “A number plus 12 is 30.”

Write and solve: “5 times a number is 65.”

A taxi charges $4 plus $2 per km. A ride cost $30. How far was it?

A square has a perimeter of 48 cm. Write an equation and find the side length.

Two numbers add to 50. One is 14 more than the other. Find them.

More practice: on your own

Snow company A: $200 per season plus $10 per visit. Company B: $30 per visit. Find the break-even point, then say which is cheaper for 15 visits.

An isosceles triangle has a perimeter of 50 cm. Its base is 4 cm shorter than each of the two equal sides. Find all three sides.

A jar holds 40 coins, all loonies ($1) and toonies ($2), worth $62. How many of each?

Marks of 72, 85 and 90 are on a report. What mark on the fourth test gives an average of 84?

Three consecutive even integers add to 90. Find them.

Challenge

Two cyclists are 60 km apart on a trail and ride toward each other, one at 18 km/h and the other at 12 km/h. How long until they meet?

Trail mix is made from nuts at $16/kg and raisins at $6/kg. How much of each makes 10 kg of mix worth $10/kg?

Write a real-life problem that can be modelled by 3(x + 4) = 2x + 20. Solve it and verify in your context.

Check your understanding

Quiz

What is the best first written step when creating an equation for a word problem?

A rectangle's length is twice its width, and its perimeter is 36 cm. Which equation fits?

Plan A costs $10 plus $2 per hour. Plan B costs $4 per hour. After how many hours are they equal?

Three consecutive integers add to 48. What are they?

Two angles add to 180°. One is 3 times the other. What is the smaller angle?

What does it mean to verify a solution in context?

For parents and teachers

Big idea: C1.5 asks students to create and solve equations for various contexts and verify their solutions. Creating the equation is usually the hard part. The six-step plan (understand, let statement, equation, solve, verify in context, answer sentence) gives structure. This lesson also builds Strand A connections to real life and careers (A2).

Common difficulties:

No let statement, so the student loses track of what x means.
Writing the second unknown with a new letter instead of in terms of the first (150 − s).
Verifying only in the equation: if the equation was set up wrong, the check will still “work”.
Answering with just a number, not a sentence that answers the question.

How to help: Do the break-even activity with a real decision your family faces. Teens find algebra more convincing when it settles a real argument.