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Solving Simultaneous Equations Using Elimination
(KS4, Year 10)
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.StepbyStep:
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.
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
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.StepbyStep:
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
x − y = 1
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...we eliminated the x's. They both have a coefficient of 1 (which does not need to be written in front of it.)
x + y = 3 
In the second example, the simultaneous equations were...
x + y = 5...we eliminated the y's. They both have a coefficient of 1 (which does not need to be written in front of it.)
x − y = 1
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 = 2yBy subtracting a negative letter, you are adding the positive letter.

Subtract the constants:
3 − 1 = 2
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