Simultaneous Equations

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What are they?

When we talk about “simultaneous equations”,  we mean two equations that intercept each other – i.e. cross through the same point (coordinates) – if you drew them on a graph.

Two equations that pass through the same point

For instance, this graph shows the two equations: y = x and y = 2x

They both intercept the origin – and therefore each other.

This is the solution: x = 0, y = 0 or (0,0) written as a coordinate.

In this case, we can see visually where they cross each other, but when we are asked to “solve” simultaneous equations – we have to do so without a graph, algebraically – here’s how.


FUN FACT To help you remember

The word “simultaneous” literally means “occurring, operating, or done at the same time.” You might simultaneously eat whilst watching TV…and you might simultaneously be texting someone whilst revising right now (but let’s face it – you can’t do both things properly at the same time!).


Solving them algebraically

We need to be able to solve simultaneous equations algebraically, without a graph, the best way to learn how to do this is to work through an example and practice yourself.

So off we go…

Solve:

2x = 6 – 4y

-3 – 3y = 4x

STEP 1: Label the equations

  2x = 6 – 4y        1

-3 – 3y = 4x         2

STEP 2: Rearrange into ax + by = c

          -4y becomes+4y                                        2x + 4y = 6            1

            +4x becomes-4x                                    – 4x – 3 – 3y = 0       2 

              -3 becomes+3                                      – 4x – 3y = 3              2

STEP 3: Multiply either or both equations to make either coefficient match

                  2x becomes 4x                             4x + 8y = 12         1 x 2

                  4x remains  4x                                – 4x – 3y = 3         2

STEP 4: Add/subtract the 2 equations to get rid of the equivalent coefficients – in this case, we have a +4x and a -4x, so we need to add them to get to zero (+4 + -4 = 0)

                                     1 + 2:                4x + 8y = 12        +          – 4x – 3y = 3

To get:             5y = 15

So:                 y = 3

But this is only half of the solution – we need the x coordinate too – remember we are finding the point where the two equations cross each other.

STEP 5: Substitute in either of the original equations

                            Substitute y = 3 into the original equation 1 : 2x = 6 – 4y    

                                                                   To get:            2x = 6 – 4(3)

2x =6 – 12

So:             x = -3

STEP 6: OUR SOLUTION

x = -3, y = 3