## 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.*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.*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

## 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

**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

**y**'s. They both have a coefficient of 1 (which does not need to be written in front of it.)

**x**in each equation is different (1 and 2). The coefficients of the

**y**in each equation is different (2 and 3).

how to solve simultaneous equations where the coefficients differ

## Top Tip

## Add or Subtract?

- If the unknown you wish to eliminate has the
**s**ame sign,**s**ubtract the equations. - If the unknown you wish to eliminate has
**d**ifferent signs, a**dd**the equations.

## Beware

## Be Careful When Subtracting Equations

Consider the simultaneous equations shown below:x + y = 3 ... (1)

x − y = 1 ... (2)

*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

- Subtract the constants:
3 − 1 = 2

**2y = 2**, from which we can solve the equations.

## You might also like...

simultaneous equationssolving simultaneous equationssolving simultaneous equations using elimination (where coefficients differ)solving simultaneous equations using substitution

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