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Lesson overview

Algebra - systems of linear equations

Simultaneous equations are two or more equations that share the same variables. We solve them by finding values that satisfy both equations at the same time. There are two main methods: elimination (adding or subtracting the equations to remove a variable) and substitution (solving one equation for a variable and replacing it in the other). Graphically, the solution is the point where the two lines intersect. For example, the equations $x + y = 5$ and $2x - y = 1$ have the solution $x = 2, y = 3$ because when $x = 2$ and $y = 3$ , both equations are true.

Solving by elimination

Solve $2x + y = 7$ and $x + y = 5$ using elimination.

Subtract equation (2) from equation (1) to eliminate $y$ . This gives $x = 2$ . Substitute back to find $y$ .

Solving by substitution

Solve $y = 2x + 1$ and $3x + y = 11$ using substitution.

Since equation (1) already gives $y$ in terms of $x$ , substitute directly into equation (2).

Elimination when coefficients need adjustment

Solve $3x + 2y = 13$ and $2x + y = 8$ using elimination.

Sometimes you need to multiply one equation by a number so the coefficients of one variable match.

Checking your solution

Verify that $x = 1, y = 2$ is the solution to $x + y = 3$ and $2x + y = 4$ .

Always substitute your answer back into both original equations to check.

Worked example

Solve the simultaneous equations $4x + 3y = 17$ and $2x - 3y = 1$ using the elimination method.

Label the equations: equation (1) is $4x + 3y = 17$ and equation (2) is $2x - 3y = 1$ .
Notice that the $y$ coefficients are opposite: $+3y$ and $-3y$ . This is perfect for elimination.
Add equation (1) and equation (2): $(4x + 3y) + (2x - 3y) = 17 + 1$ .
The $3y$ and $-3y$ cancel: $6x = 18$ .
Solve for $x$ : $x = 3$ .
Substitute $x = 3$ into equation (1): $4(3) + 3y = 17$ , so $12 + 3y = 17$ .
Solve for $y$ : $3y = 5$ , so $y = \frac{5}{3}$ .
The solution is $x = 3, y = \frac{5}{3}$ .

Try it

For each pair of equations, choose whether elimination or substitution is easiest, then solve carefully. Always check your answer.

Question 1

Solve $x + y = 6$ and $x - y = 2$ .

💡 Add the two equations to eliminate $y$ : $(x + y) + (x - y) = 6 + 2$ gives $2x = 8$ .

Question 2

Solve $2x + y = 9$ and $x + y = 6$ .

💡 Subtract the second equation from the first to eliminate $y$ : $(2x + y) - (x + y) = 9 - 6$ .

Question 3

Solve $y = 2x$ and $x + y = 9$ .

💡 Use substitution. Replace $y$ with $2x$ in the second equation: $x + 2x = 9$ .

Question 4

Solve $3x + 2y = 12$ and $x + 2y = 8$ .

💡 Subtract the second equation from the first to eliminate $2y$ : $(3x + 2y) - (x + 2y) = 12 - 8$ .

Question 5

Solve $2x + 3y = 13$ and $4x - y = 5$ .

💡 Multiply the second equation by 3 to get $12x - 3y = 15$ , then add to the first equation.

Common mistakes

Watch for these when working through the lesson.

Forgetting to apply the same operation to both sides when multiplying equations. If you multiply equation (1) by 2, multiply all terms.
Making arithmetic errors when adding or subtracting equations. Work carefully through each step.
Not substituting correctly. When you find one variable, substitute the value (not the expression) into the other equation.
Stopping after finding one variable. Always substitute back to find the second variable and check both answers in both original equations.

Simultaneous equations

Lesson overview

Worked example

Try it

Common mistakes

Related topics

Independent practice will plug in here