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GCSE: Simultaneous equations: Part 1

Solving them algebraically

 

This particular section requires a fairly sound knowledge of solving linear equations.

To understand the geometrical interpretation, go to the section on ‘Simultaneous equations graphically’

The solutions to these equations are positive, integers (whole numbers). Part 2 will go on to develop skills involving fractions and negative solutions

Contents

1)           Solving linear simultaneous equations with positive coefficients

2)          Solving linear simultaneous equations with negative coefficients

3)          Solving linear simultaneous equations  which require you to pre-multiply at least one of the equations

 


 

Solving linear simultaneous equations

 

 

 

 becomes

 

 

 

The solution is therefore

 

 

Observe what happens if we do .

 

 

 

 

Now substitute

 

 

 

 

 

 becomes

 

 

 

So the solution is:

 

If we do

 

 

 

 

Substitute  into (2)

 

 


 

 

 

 

So, becomes

 

 

 

The solution is therefore

 

This time, if we do

 

 

 

 

Substitute  into 

 

 

 

 

So, becomes

 

 

 

The solution is therefore

 

This time, if we do

 

 

 

 

Substitute  into 

 

 


 

Sometimes we need to pre-multiply one of the equations first.

 

 

 

 

 

 

 

 

 

So, becomes

 

 

 

The solution is therefore

 

The coefficients of the y in both equations do not match. So we pre-multiply the whole of equation by 2 and call this new equation . We keep equation the same

 

 

We now do

 

 

 

 

Substitute  into 

 

 


 

 

 

 

 

 

 

 

 

 

So, becomes

 

 

 

The solution is therefore

 

The coefficients of the y in both equations do not match. So we pre-multiply the whole of equation by 5 and call this new equation .

Also we pre-multiply the whole of equation by 2 and call this new equation .

 

We now do

 

 

 

 

Substitute  into