Solve linear systems by substitution or elimination

Ontario Grade 10 Mathematics

Modelling Linear Relations

Find the values that make two linear equations true at the same time.

A linear equation can have many solutions. For example, different pairs of numbers can make

x+y=8

true. A linear system is a pair of equations that use the same variables. Its solution is a pair of values that makes both equations true. In this lesson, you will solve systems using substitution or elimination. These methods build on familiar skills: solving an equation, keeping both sides balanced, and combining like terms.

What you will learn

Explain what a solution to a linear system means.
Choose substitution or elimination to solve a pair of linear equations.
Check a solution in both original equations.
Recognize when a pair of equations has no solution or many solutions.

Grade 9 bridge: what does a solution mean?

Start with the equation

x+y=8

. The pair

(3,5)

is a solution because

3+5=8

. The pair

(2,4)

is not, because

2+4

is not

8

.

Now consider two equations together:

x+y=8

and

x-y=2

. A solution must work in both. The pair

(5,3)

works:

5+3=8

and

5-3=2

.

The variables are the unknown values. An ordered pair, such as

(5,3)

, lists the value of

x

first and the value of

y

second. A linear system is a set of linear equations with the same variables. Solving the system means finding the values that satisfy every equation in it.

Check a proposed solution in each equation, not just one.
In $(a,b)$ , the first number is the value of $x$ and the second is the value of $y$ .

Two methods for solving a system

Substitution means replacing a variable with an equal expression. It is useful when one equation already gives a variable by itself, or when that is easy to arrange. For example, if

y=2x+1

, you can replace

y

with

2x+1

in the other equation. This leaves one equation with one variable.

Elimination means combining equations so one variable disappears. You may add or subtract the equations. If a variable has matching coefficients with opposite signs, adding can remove it. If the coefficients match with the same sign, subtracting can remove it. A coefficient is the number multiplying a variable; in

3x

, the coefficient is

3

.

Whichever method you use, keep equations balanced. If you multiply an equation by a number, multiply every term on both sides. Once you find one variable, put its value into an original equation to find the other. Then check both values in both original equations.

\begin{aligned} ax+by&=c\\ dx+ey&=f \end{aligned}

Use substitution when a variable is already isolated or easy to isolate.
Use elimination when a variable can be cancelled by adding or subtracting the equations.
The final check must use the original system.

A table view: what does the solution represent?

Each equation describes pairs of values that make it true. A system asks for a pair that belongs to both sets of pairs. The table shows this idea for a simple system. The shared pair is the solution.

The methods in this lesson find the shared pair using equations. They do not depend on guessing values from a table.

A solution is shared by both equations.
A table can help check a candidate pair, but substitution or elimination gives a systematic way to find it.

Choosing a method and interpreting the result

After solving, you may find one pair that works in both equations. Some systems instead lead to a statement that is never true, such as

0=5

. That means there is no solution: no pair can satisfy both equations.

A different system may reduce to a statement that is always true, such as

0=0

. This means the equations describe the same relationship, so there are many solutions. In that case, every pair that works in one equation also works in the other.

These outcomes come from simplifying the given equations. They do not change the main test: a solution must make both original equations true.

A true numerical statement such as $0=0$ can indicate many solutions.
A false numerical statement such as $0=5$ indicates no solution.
A single ordered pair is the answer when the system has one shared pair.

Checking candidate pairs for the system $x+y=8$ and $x-y=2$

Candidate pair	First equation	Second equation	Works in both?
$(5,3)$	$5+3=8$	$5-3=2$	Yes
$(4,4)$	$4+4=8$	$4-4=0$	No

Worked example

Solve by elimination

Solve the system

2x+y=11

and

x-y=1

.

Add to remove a variable
The $y$ terms have opposite signs. Add the equations vertically. The $y$ terms cancel, leaving an equation in $x$ . \begin{aligned}2x+y&=11\\x-y&=1\\\hline 3x&=12\end{aligned}
Find the value of x
Divide both sides by $3$ to keep the equation balanced. This gives the value of $x$ .
$x=4$
Find the value of y
Substitute $x=4$ into the original equation $x-y=1$ . Solve for $y$ .
$4-y=1 \Rightarrow y=3$
Check both equations
Replace $x$ with $4$ and $y$ with $3$ in each original equation. Both statements are true, so the pair is the solution.
$2(4)+3=11 4-3=1$

Answer: The solution is

(4,3)

.

Check: The pair makes both original equations true.

Common mistakes and how to avoid them

Stopping after finding one variable.

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

Changing only one term when multiplying an equation.

Correction: Multiply every term on both sides by the same number so the equation stays balanced.

Checking the answer in only one equation.

Correction: Substitute the ordered pair into both original equations. Both must be true.

Thinking

0=0

means there is no solution.

Correction: It is a true statement and can show that the equations describe the same relationship, giving many solutions.

Lesson summary

A system has two or more equations using the same variables.
Substitution replaces a variable with an equal expression.
Elimination adds or subtracts equations to cancel a variable.
Use an original equation to find the second variable, then check both equations.
A false statement after simplifying means no solution; an always-true statement can mean many solutions.

Check your understanding

Question 1

For the system

x+y=9

and

x-y=3

, what is the solution?

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

Show answer and explanation

(6,3)

For

(6,3)

, the equations give

6+3=9

and

6-3=3

. The other pairs do not satisfy both equations.

Question 2

In the system

y=2x+1

and

x+y=10

, which substitution gives an equation with one variable?

Replace $y$ in the second equation with $2x+1$ .
Replace $x$ in the first equation with $2x+1$ .
Add $y$ to both equations.
Replace $2x+1$ with $10$ .

Show answer and explanation

Replace

y

in the second equation with

2x+1

.

Since the first equation states that

y

equals

2x+1

, replace

y

in the other equation. This gives

x+(2x+1)=10

.

Question 3

A system simplifies to

0=5

. What does this tell you?

The system has no solution.
The system has one solution, $(0,5)$ .
The system has many solutions.
The value of each variable is $5$ .

Show answer and explanation

The system has no solution.

The statement

0=5

is false. The original equations cannot both be true for any pair of values.

Key terms

Linear system: A group of linear equations that use the same variables and must be true at the same time.
Solution: The value or values that make every equation in a system true.
Substitution: A method that replaces a variable with an equal expression.
Elimination: A method that combines equations to cancel one variable.
Coefficient: The number multiplying a variable in a term.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic G12. 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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