Model and solve real situations with linear systems

Ontario Grade 10 Mathematics

Analytic Geometry

Turn two connected conditions into equations, then find values that satisfy both.

You already know how to use a variable for an unknown number and solve an equation by keeping both sides balanced. Some real situations have two unknown amounts and two pieces of information. For example, you may know both the total number of tickets sold and the total money collected. A linear system is a set of equations that use the same variables. Its solution is a set of values that makes every equation true. In this lesson, you will define the unknowns, build a system from a situation, solve it, and explain what the answer means.

What you will learn

Identify two unknown quantities and represent them with variables.
Write two linear equations from the conditions in a real situation.
Solve a pair of linear equations using substitution or elimination.
Check that a solution makes sense in the original situation.

1. Grade 9 bridge: name the unknowns and model the conditions

A variable is a letter that stands for an unknown number. Before writing an equation, decide what each variable represents. State its meaning clearly. For example, you could let

a

represent the number of adult tickets and

y

represent the number of youth tickets.

A linear equation is an equation in which a variable is not raised to a power or multiplied by another variable. A linear system is two or more linear equations considered together because they use the same variables. Each equation represents a condition from the situation.

Look for two different facts. A total count usually comes from adding quantities. A total cost comes from multiplying each count by its price, then adding the amounts. These facts are not interchangeable. If there are

a

adult tickets and

y

youth tickets, the total count is

a+y

. If adult tickets cost CAD 8 and youth tickets cost CAD 5, the total money is represented by

8a+5y

.

Each variable has a meaning and often a unit. Ticket counts are measured in tickets. A cost is measured in dollars. Keeping track of these meanings helps you choose the right equation and judge whether an answer is sensible.

\begin{cases}a+y=13\\8a+5y=86\end{cases}

Define each variable before using it.
Translate each separate condition into an equation.
A solution must make every equation in the system true.

2. Solve the system with elimination or substitution

Elimination means adding or subtracting equations so that one variable disappears. A coefficient is the number multiplying a variable. If the coefficients of one variable are opposites, adding the equations removes that variable. If the coefficients are equal, subtracting one equation from the other removes it.

When coefficients do not match, you can multiply every term on both sides of an equation by the same number. This keeps the equation balanced. Choose a multiplier that makes one variable’s coefficients equal or opposite. Then add or subtract the equations.

Substitution is another method. It means replacing a variable with an equal expression from another equation. For example, if one equation tells you that

y=13-a

, you can replace

y

with

13-a

in the other equation. This is especially useful when a variable is already by itself or is easy to isolate.

After finding one variable, use either original equation to find the other. Then check both values in both original equations. A pair of values that works in only one equation is not a solution to the system.

\text{solution}=(a,y)

Use elimination when adding or subtracting can remove a variable.
Use substitution when an equation gives one variable in terms of the other.
When multiplying an equation, multiply every term on both sides.

3. Interpret the answer and practise modelling

A real-world answer needs more than two numbers. Say what each number represents and include units when they help. Then check that the values fit the situation. Counts of tickets or people should be whole numbers and cannot be negative.

Independent practice: A school sells regular meal tickets for CAD 9 and reduced-price tickets for CAD 6. It sells 15 tickets and collects CAD 111. Let

r

be the number of regular tickets and

d

be the number of reduced-price tickets. Write one equation for the total count and one for the total money. Solve the system, then check both conditions.

For this practice situation, the count condition is

r+d=15

and the money condition is

9r+6d=111

. Try solving before reading the worked example. You can use elimination by making the coefficients of one variable match, or substitution by writing one variable in terms of the other.

A useful routine is: define the unknowns, write an equation for each condition, solve, check in the original equations, and answer in words. If the check fails, review the variable meanings, the model, and the arithmetic.

Distinguish a total count from a total cost.
Check the values in the original conditions.
Finish with a sentence that explains the values in context.

Connect each condition to its equation

Condition	What to calculate	Equation
13 tickets altogether	Add the adult and youth counts	$a+y=13$
CAD 86 collected	Multiply each count by its price, then add	$8a+5y=86$

Worked example

Find the numbers of two ticket types

A community event sells adult tickets for CAD 8 each and youth tickets for CAD 5 each. It sells 13 tickets and collects CAD 86. How many of each type are sold?

Define the unknowns
Let $a$ be the number of adult tickets and $y$ be the number of youth tickets. Both variables represent ticket counts.
Write the equations
The counts add to 13. For the money condition, multiply each ticket count by its price and add the results. This gives a pair of equations using the same unknowns. \begin{cases}a+y=13\\8a+5y=86\end{cases}
Make the coefficients match
Multiply every term in the count equation by 5. The coefficient of $y$ is now 5, matching the coefficient of $y$ in the money equation.
$5a+5y=65$
Eliminate a variable
Subtract the new equation from the money equation. The $y$ terms cancel, leaving an equation for $a$ . \begin{aligned}(8a+5y)-(5a+5y)&=86-65\\3a&=21\end{aligned}
Find the other value
Divide by 3 to get the adult-ticket count. Then use the total-count equation to find the youth-ticket count.
$a=7, y=13-7=6$
Check and interpret
The counts add to 13. The adult tickets bring in CAD 56, and the youth tickets bring in CAD 30. Together they bring in CAD 86, so both conditions are satisfied.
$7+6=13, 8(7)+5(6)=86$

Answer: The event sells 7 adult tickets and 6 youth tickets.

Check: Both original conditions work: there are 13 tickets, and their total revenue is CAD 86. The values are whole, non-negative ticket counts.

Common mistakes and how to avoid them

Using the total number of tickets as the money equation.

Correction: Write a separate equation for each condition. For money, multiply each ticket count by its own price.

Multiplying only one term in an equation by a number.

Correction: Multiply every term on both sides by the same number to keep the equation balanced.

Stopping after finding one variable.

Correction: Use that value in an original equation to find the other variable, then check both conditions.

Giving two numbers without saying what they represent.

Correction: State which value belongs to each variable and connect the values to the situation.

Lesson summary

A linear system uses equations with the same unknowns to represent connected conditions.
Define variables clearly and write a separate equation for each condition.
Use substitution or elimination to find values that satisfy every equation.
Check the result in the original situation and explain what the values mean.

Check your understanding

Question 1

A fundraiser sells adult tickets for CAD 10 and child tickets for CAD 4. What does the equation

10a+4c=76

represent, if

a

and

c

are the numbers of adult and child tickets?

There are 76 tickets altogether.
The ticket sales bring in CAD 76 altogether.
Each adult ticket costs CAD 76.
There are 10 adult tickets and 4 child tickets.

Show answer and explanation

The ticket sales bring in CAD 76 altogether.

The terms

10a

and

4c

represent the money from adult and child tickets. Their sum is the total revenue.

Question 2

For the system

x+y=10

and

2x+y=14

, what is the solution?

$(x,y)=(4,6)$
$(x,y)=(6,4)$
$(x,y)=(7,3)$
$(x,y)=(3,7)$

Show answer and explanation

(x,y)=(4,6)

Subtract the first equation from the second to get

x=4

. Then the first equation gives

y=6

. Thus, the solution is

(4,6)

.

Question 3

Why should a solution to a ticket system be checked in the original equations?

To confirm the values satisfy both the count and cost conditions.
To make the variables have different meanings.
To change the ticket prices until the values are whole numbers.
To avoid writing a final answer.

Show answer and explanation

To confirm the values satisfy both the count and cost conditions.

A system’s solution must satisfy both equations. Checking confirms that both the item count and total cost work.

Key terms

Variable: A letter that stands for an unknown or changing number.
Linear equation: An equation in which variables are not raised to powers and are not multiplied by each other.
Linear system: Two or more linear equations considered together because they use the same variables.
Coefficient: The number multiplying a variable.
Elimination: A method that adds or subtracts equations to make one variable disappear.
Substitution: A method that replaces a variable with an equal expression from another equation.
Solution: The value or values that make every equation in a system true.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic G5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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