Solve two-variable linear systems by substitution or elimination

Ontario Grade 10 Mathematics

Analytic Geometry

Use substitution or elimination to find the pair that makes both equations true.

A linear equation with two variables can have many solutions. For example, different pairs of numbers may make an equation true. A system is a group of equations that must be true at the same time. In this lesson, you will find the ordered pair that works in both equations. You will use substitution or elimination, then check your answer in the original equations.

What you will learn

Explain what a solution to a two-variable linear system represents.
Solve a system by substitution.
Solve a system by elimination.
Check a solution in both original equations.

1. Grade 9 bridge: equations and ordered pairs

An equation says that two expressions have the same value. You can keep an equation balanced by doing the same operation to both sides. For example, if you add the same number to each side, the equality stays true.

A variable is a letter that stands for a number. In a two-variable equation, such as

x+y=10

, both

x

and

y

stand for numbers. An ordered pair, written

(x,y)

, gives a value for each variable in that order.

A solution to a system is one ordered pair that makes every equation in the system true. The two equations might describe two number relationships. The solution is the pair that satisfies both relationships.

Keep the order of values consistent: the first value is for $x$ , and the second is for $y$ .
A pair that works in only one equation is not enough.

2. What substitution and elimination do

Substitution means replacing a variable with an equal expression. First, isolate one variable in one equation. Isolate means to get that variable by itself. Then replace that variable in the other equation with the expression that equals it. This leaves an equation with one variable.

For example, if one equation tells you

y=8-x

, then every place you see

y

in the other equation can be replaced with

8-x

. The replacement is valid because the two expressions have the same value.

Elimination means combining equations so that one variable disappears. To eliminate a variable, make its coefficients opposites, then add the equations. A coefficient is the number multiplying a variable. For example, in

3x+2y=14

, the coefficient of

y

is

2

.

If the coefficients of a variable are already opposites, add the equations directly. If they are not, multiply one or both entire equations by a number first. Multiply every term on both sides so the equation remains balanced. After one variable disappears, solve the remaining equation, then substitute the value back to find the other variable.

Choose substitution when a variable is already alone or can be isolated easily. Choose elimination when a variable’s coefficients are equal or can be made opposites without much work. Both methods aim to reduce the system to one equation with one variable.

Substitution replaces a variable with an equal expression.
Elimination combines equations to cancel one variable.
After finding one variable, use an original equation to find the other.

3. Guided example: use both methods and check

A school club sells adult tickets for CAD 8 and student tickets for CAD 5. One order contains 7 tickets and costs CAD 47. Let

a

be the number of adult tickets and

s

the number of student tickets. The total number of tickets gives one equation. The total cost gives another.

We can solve this system by substitution. The ticket-count equation makes it easy to express one variable in terms of the other. We can also use elimination because the

s

terms have opposite coefficients after we write the equations in standard order.

In the steps below, both methods lead to the same pair. The final check puts the pair into the original equations, not just the rearranged equations.

a+s=7,\quad 8a+5s=47

The variables describe ticket counts, while the equations represent ticket count and total cost.
A correct answer must satisfy both conditions.

4. Independent practice and a reliable check

Try these systems on your own. For the first one, use substitution because one variable is already isolated. For the second, use elimination because the

y

coefficients are opposites. Write each solution as an ordered pair and check it in both equations.

System A:

y=2x+1

and

x+y=10

. System B:

3x+2y=16

and

x-2y=0

. These are practice prompts, not additional worked examples.

A check is simple: replace each variable in each original equation with your proposed value, then calculate both sides. The left side and right side must be equal in both equations. If either equation fails, review the signs, arithmetic, and substitution.

System A has solution $(3,7)$ .
System B has solution $(4,2)$ .
Use both original equations when checking.

What each equation tells us

Condition	Equation	Meaning
Ticket count	$a+s=7$	The two ticket types total 7.
Ticket cost	$8a+5s=47$	The total cost is CAD 47.
Solution	$(a,s)=(4,3)$	Both conditions are true.

Worked example

Ticket sales system

Solve

a+s=7

and

8a+5s=47

. Here,

a

is the number of adult tickets and

s

is the number of student tickets.

Isolate a variable
The first equation is easy to rearrange. Subtract $a$ from both sides to express $s$ by itself.
$s=7-a$
Substitute
Replace $s$ in the cost equation with $7-a$ . This is valid because the first equation says these expressions have the same value.
$8a+5(7-a)=47$
Solve for a
Distribute $5$ , combine like terms, and then isolate $a$ using the same operation on both sides.
$8a+35-5a=47\quad\Rightarrow\quad 3a=12\quad\Rightarrow\quad a=4$
Find s
Replace $a$ with $4$ in the equation $s=7-a$ . This gives the number of student tickets.
$s=7-4=3$
Verify by elimination
The same system can be written with the terms in matching order. Add the equations to cancel $s$ . The result agrees with $a=4$ .
$\begin{aligned}a+s&=7\\8a+5s&=47\\8a+5s&=47\end{aligned}$

Answer: There are 4 adult tickets and 3 student tickets, so the solution is

(a,s)=(4,3)

.

Check: The count is

4+3=7

. The cost is

8(4)+5(3)=32+15=47

. Both original equations are true.

Common mistakes and how to avoid them

Changing a sign while rearranging an equation.

Correction: Use the same operation on both sides. For example, subtract

a

from both sides of

a+s=7

to get

s=7-a

.

Multiplying only one term when preparing for elimination.

Correction: Multiply every term on both sides of the equation by the same number.

Stopping after finding only one variable.

Correction: Substitute that value into an original equation to find the second variable.

Checking the answer in just one equation.

Correction: Substitute the ordered pair into both original equations. A solution must make both true.

Lesson summary

A system’s solution is an ordered pair that makes both equations true.
Use substitution to replace a variable with an equal expression.
Use elimination to combine equations and cancel a variable.
Find the second variable after solving for the first.
Check the pair in both original equations.

Check your understanding

Question 1

Which ordered pair solves

x+y=9

and

x-y=3

?

$(6,3)$
$(3,6)$
$(9,3)$
$(5,4)$

Show answer and explanation

(6,3)

For

(6,3)

, the sum is

9

and the difference is

3

. The other pairs do not make both equations true.

Question 2

In the system

y=4-x

and

2x+y=11

, what should replace

y

in the second equation when using substitution?

$4-x$
$4+x$
$11-2x$
$x-4$

Show answer and explanation

4-x

The first equation states that

y

equals

4-x

, so that expression replaces

y

.

Question 3

What is the main purpose of adding two equations in elimination?

To make one variable cancel
To make both variables disappear
To change the solution
To turn the equations into inequalities

Show answer and explanation

To make one variable cancel

When the coefficients of one variable are opposites, adding the equations makes that variable’s total coefficient zero.

Key terms

System: A group of equations that must be true at the same time.
Ordered pair: A pair of values written in order, usually $(x,y)$ , that gives values for two variables.
Substitution: Replacing a variable with an expression that has the same value.
Elimination: Combining equations so that one variable cancels.
Coefficient: The number multiplying a variable in a term.

Continue through MPM2D

View the complete MPM2D Ontario Grade 10 Mathematics curriculum and lessons

Tutor support for G4 · Solve two-variable linear systems by substitution or elimination

Working on G4 · Solve two-variable linear systems by substitution or elimination in MPM2D? Bring the question and your own attempt. In a live session, you can trace a small input step by step, compare expected and actual output, and test an edge case. Ask about the first step that does not make sense, then try a similar question yourself.

Explore math tutoring · Request an eligible free 30-minute introduction. Your request is matched by subject and language; a tutor must accept before a time is confirmed.

برای مبحث G4 · حل دستگاه دو معادله خطی با جایگذاری یا حذف در درس MPM2D، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید یک ورودی کوچک را گام‌به‌گام دنبال کنید، خروجی مورد انتظار و واقعی را مقایسه کنید و یک حالت مرزی را بیازمایید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic G4. 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.

Official curriculum reference

Report a correction or ask a question