If we have 2 equations with 2 unknown variables, there is only one solution set that satisfies both equations.

Algebraically, there are two methods for finding this solution:

The Method of Elimination to solve the simualtaneous equations– this is widely regarded as the simpler of the two methods, and the most commonly taught, both equations must be linear to use this method.

Elimination involves combining the two equations together to make one equation in terms of only one variable, which we can solve easily.

How?

1. Prepare the two equations such that if you add or subtract the two equations together one variable cancels out.

In other words, you want to multiply or divide an equation (or both) so that the number (not necessarily the ± sign) next to the variable in both equations is the same.

2. You add/subtract the equations to form one equation with a one variable to solve for this variable.

3. You plug the answer you got in 2, back in to one of your original equations, to find the other variable.

4. Check these are correct by plugging both your answers into the other equation.

 

For example,

a) 2y = x + 2
b) y = 16 - 2x 

1. We have two easy options here:

We can either multiply equation a) by 2, which gives you 

a*) 4y = 2x + 4

which means the number multipled by x, 2, is the same as in b) (remember the ± sign doesn't matter here)

or we can multiply equation b) by 2, which gives you 

b*) 2y = 32 - 4x

which means the number multipled by y, 2, is now the same as in a).

Lets carry on with our first option and a*.

2.

The equations we are now working with are

a*) 4y = 2x + 4
b) y = 16 - 2x 

If we add these 2 equations together, the x variable gets eliminated (hence the name):

a*+b : 5y = 20

Remember to keep each term on the correct side, add up terms on the left together, add up terms on the right together. 

We can no solve 5y = 20 easily! Just divide both sides by 5.

y = 4

3. Plug this y value into one of the original equations, lets take a). 

a) 2y = x + 2

Using y = 4

2(4) = 8 = x + 2 
x = 6

4. Lets check this combination first with our second original equation b).

b) y = 16 - 2x 

16 - 2(6) = 16 -12 = 4 = y

It works.

We are done.

So x = 6, y = 4.

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