The Lesson

Simultaneous equations are a set of several equations with several unknowns. We can use the elimination method to find the values of the unknowns which solve both equations at the same time.

How to Solve Simultaneous Equations Using the Elimination Method

Solving simultaneous equations using the elimination method is easy.

Question

Solve the simultaneous equations shown below using the elimination method.
We can eliminate the x if we subtract Equation (2) from Equation (1).

Step-by-Step:

1

Subtract Equation (2) from Equation (1). Write the equations in a subtraction.

2

Subtract the x terms.
x − x = 0
We have eliminated the x.

3

Subtract the y terms.
2y − y = y

4

Subtract the constants.
5 − 3 = 2

5

We have eliminated one unknown (the x). Solve for y.

y = 2 is a solution to the simultaneous equations.

6

Substitute the variable we have just found (y = 2) into one of the equations. Solve for x.

x + 2y = 5

x + 2( 2 ) = 5

x + 2 × 2 = 5

x + 4 = 5

x + 4 − 4 = 5 − 4

x = 1

x = 1 is a solution to the simultaneous equations.

Answer:

We have solved the simultaneous equations:

x = 1, y = 2 solves

x + 2y = 5

x + y = 3

A Real Example of How to Solve Simultaneous Equations Using the Elimination Method

Question

Solve the simultaneous equations shown below using the elimination method.
We can eliminate the y if we add Equation (1) to Equation (2).

Step-by-Step:

1

Add Equation (1) to Equation (2). Write the equations in an addition.

2

Add the x terms.
x + x = 2x

3

Add the y terms. Don't forget: Adding a negative letter is the same as subtracting the (positive) letter.
y + −y = 0
We have eliminated the y.

4

Add the constants.
5 + 1 = 6

5

We have eliminated one unknown (the y). Solve for x.

2x = 6

2x ÷ 2 = 6 ÷ 2

x = 3

x = 3 is a solution to the simultaneous equations.

6

Substitute the variable we have just found (x = 3) into one of the equations. Solve for y.

x + y = 5

3 + y = 5

3 + y − 3 = 5 − 3

y = 2

y = 2 is a solution to the simultaneous equations.

Answer:

We have solved the simultaneous equations:

x = 3, y = 2 solves

x + y = 5

x − y = 1

Lesson Slides

The slider below shows another real example of how to solve simultaneous equations using the elimination method.

Solving Simultaneous Equations Using Elimination Where Coefficients of Variables Differ

In the examples on this page, the coefficients of the variable we have eliminated have been the same in both equations.
  • In the first example, the simultaneous equations were...

    x + 2y = 5

    x + y = 3

    ...we eliminated the x's. They both have a coefficient of 1 (which does not need to be written in front of it.)
  • In the second example, the simultaneous equations were...

    x + y = 5

    x − y = 1

    ...we eliminated the y's. They both have a coefficient of 1 (which does not need to be written in front of it.)
In many simultaneous equations, the unknowns will not have the same coefficients. The coefficients of the x in each equation is different (1 and 2). The coefficients of the y in each equation is different (2 and 3).
Read more about how to solve simultaneous equations where the coefficients differ

Top Tip

Add or Subtract?

  • If the unknown you wish to eliminate has the same sign, subtract the equations.
  • If the unknown you wish to eliminate has different signs, add the equations.

Beware

Be Careful When Subtracting Equations

Consider the simultaneous equations shown below:

x + y = 3 ... (1)

x − y = 1 ... (2)

You decide to eliminate the x by subtracting Equation (2) from Equation (1).
  • The x's cancel:
    x − x = 0
  • Be careful with the y terms:

    y − (−y) = y −− y

    y − (−y) = y + y = 2y

    By subtracting a negative letter, you are adding the positive letter.
  • Subtract the constants:
    3 − 1 = 2
By subtracting the equations, we find 2y = 2, from which we can solve the equations.