# How to Solve Simultaneous Equations Using the Elimination Method

The elimination method is used to solve simultaneous equations.

It involves adding or subtracting simultaneous equations (or multiples of them) to eliminate one of the unknowns. This leaves one equation with one unknown, which can be solved to find the unknown.

The elimination method varies whether the unknowns (such as x and y) have:

• The same or opposite signs.
• The same or different numbers in front of them.
Note: Another way of saying the above is whether the unknowns have the same (sign and absolute value).

This means there are four cases, which will be looked at separately:

Same sign   Different sign
Same number   Case 1   Case 2
Different number   Case 3   Case 4

General Approach to Eliminating an Unknown:
• Decide which unknown (x or y) you wish to eliminate.
• Ensure this unknown has the same number in front of it, if necessary by multiplying one or both equations.
• If the unknowns have the same sign, subtract the equations.
• If the unknowns have different signs, add the equations
This approach will be used in each of the 4 cases to eliminate one unknown.

Note: Elimination leaves one equation with one unknown, which can be solved to find the unknown which was not eliminated. The other unknown can then be found. This step is common to each of the 4 cases, and will be shown once, separately.

# Using the Elimination Method When the Unkwowns Have the Same Signs and Same Number (Case 1)

Question: What values of x and y solve the simultaneous equations below?

In this case, the of x are both positive and have the same number.

Eliminate the x by subracting Equation (2) from Equation (1).

The subtraction sign applies to the whole of Equation (2).

The best way to do the subtraction is to draw an equals line underneath the 2 equations, and then subtract down each column - subtracting the x terms, then the y terms, then the constant terms on the right hand side of the equals sign:

Note that the x term has disappeared. This is the whole point of the elimination method - one unknown has been eliminated.

One equation in one unknown is left which can be solved to find that unkwown. This value can then be substituted into one of the original simultaneous equations to find the other unknown. The simultaneous equations have then been solved. (See 'Solving Simultaneous Equations Once an Unknown Has Been Eliminated' below)

The solution to the simultaneous equation is:

# Using the Elimination Method When the Unkwowns Have Different Signs and the Same Number (Case 2)

Question: What values of x and y solve the simultaneous equations below?

In this case, the of y are different - they have the same number but one is positive and one is negative.

Eliminate the x by adding Equation (2) to Equation (1).

The addition sign applies to the whole of Equation (2).

The best way to do the subtraction is to draw an equals line underneath the 2 equations, and then add up each column - adding the x terms, then the y terms, then the constant terms on the right hand side of the equals sign:

Note that the x term has disappeared. This is the whole point of the elimination method - one unknown has been eliminated.

One equation in one unknown is left which can be solved to find that unkwown. This value can then be substituted into one of the original simultaneous equations to find the other unknown. The simultaneous equations have then been solved. (See 'Solving Simultaneous Equations Once an Unknown Has Been Eliminated' below)

The solution to the simultaneous equation is:

# Using the Elimination Method When the Unkwowns Have Different Numbers and the Same Sign (Case 3)

Question: What values of x and y solve the simultaneous equations below?

In this case, the of x are different - they have different numbers but the same signs.

Firstly, the numbers in front of the x terms in both equations need to be made the same. This is done by multiplying one or both equations.

Multiply Equation (1) by 1 (leaving it the same) and multiply Equation (2) by 2. (See the Note 'How to make the the same', right):

This is simply Case 1, where the unknown we wish to eliminate has the same number and sign in front of it.

As the signs are the same, subtract the equations to eliminate x, and eventually to find the solution:

# Using the Elimination Method When the Unkwowns Have Different Numbers and Different Signs (Case 4)

Question: What values of x and y solve the simultaneous equations below?

In this case, the of y are different - they have different numbers and different signs.

Firstly, the numbers in front of the y terms in both equations need to be made the same. This is done by multiplying one or both equations.

Multiply Equation (1) by 3 and multiply Equation (2) by 2. (See the Note 'How to make the the same', right):

This is simply Case 2, where the unknown we wish to eliminate has the same number in front of it but different signs.

As the signs are different, add the equations to eliminate y, and eventually to find the solution:

# Solving Simultaneous Equations Once an Unknown Has Been Eliminated

In each of the 4 cases above, once the equations were added or subtracted, one equation was left with one unknown.

In Case 1, the equation left was:

This is an algebraic equation that contains multiplication.

The x is multiplied by 2, so divide both sides by 2 to find x:

y = 2.

Substitute y = 2 into either of the original simultaneous equations to find x.

Substituting y = 2 into Equation (1) from Case 1:

This is an algebraic equation that contains addition.

2 is added to x, so subtract 2 from both sides to find x:

x = 3.

The solution to the simultaneous equations is:

The slider below shows an example of solving simultaneous equations using the elimination method:
show

##### Top Tip

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.

##### Note
WHAT ARE COEFFICIENTS?

The of x and y are the numbers in front of the x and y in the equation.

Note: When x or y appear on their own, without a number in front of it, its is 1.

The sign is also part of the co-efficent, as well as the absolute value, so the co-efficent of the y above is -1.

# CAREFUL WHEN SUBTRACTING EQUATIONS

Note the subtraction of the y terms highlighted in yellow to the left.

The subtraction is y - (-y). Subtracting a negative number is equivalent to adding. So, y - (-y) = y + y = 2y.

##### Note
HOW TO MAKE THE COEFFICIENTS THE SAME

If the of the unknown you wish to eliminate are not the same, they need to be made the same by multiplying one or both equations.

The new will be a common multiple of the original co-efficients.

One way of ensuring the co-efficents the same is to:

• multiply Equation (1) by the of x in Equation (2)
• multiply Equation (2) by the of x in Equation (1)
Using the example given to the left:

• The of x in Equation (2) is 4. Multiply Equation (2) by 4.
• The of x in Equation (1) is 2. Multiply Equation (1) by 2.

Both are 8, and so can be eliminated (by subtraction in this case as the signs are the same).

While this method guarantees a new that is a common multiple of the original co-efficients, it is not always the quickest way.

In the example given, 4 is also a common multiple of the original co-efficients, 4 and 2.

In which case, there is no need to multiply Equation (1) by anything, while Equation (2) should be multiplied by 2.

This requires multiplying just one equation, which is quicker than multiplying 2.

In this case, the new co-efficient, 4, is the lowest common multiple of the original co-efficients, 4 and 2.

WHAT IS THE LOWEST COMMON MULTIPLE AND HOW DO I USE IT?

A multiple of a number is the result of multiplying a number by an integer.

For example, the multiples of the co-efficients, 2 and 4, are:

The lowest common multiple is the lowest number that appears in both lists:

4 is the lowest common denominator.

To find what to multiply the equations by, look at which multiple the lowest common multiple is. This can be done by listing 1, 2 ,3 above the multiples, as below:

As higlighted in red, 4 is the 2nd multiple of 4. So Equation (2) (where the of x is 2) needs to be multiplied by 2.

As highlighted in yellow, 4 is the 1st multiple of 4. So Equation (1) (where the of x is 4) needs to be multiplied by 1.