GCSE Mathematics/Simultaneous Equations

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Solving simultaneous equations[edit]

By elimination[edit]

One way of solving a simultaneous equation is by canceling out either the x or y values so that you are left with a linear equation.

First example[edit]

20x + 15y = 135
20x - 8y = 20

In this example, we could subtract the second equation from the first to get this:

23y = 115
y = 5

Once we know this, we can go back to one of the original equations, and replace y with 5, then solve it, like this:

20x + 15(5) = 135
20x = 135 - 75
x = \frac{60}{20} = 3

So, the final solution is:

x = 3
y = 5

Second example[edit]

4x + 2y = 12
x + y = 4

We can see that in this example the equations will not cancel each other out. To make them cancel each other out, we multiply the second equation by two and get:

2x + 2y = 8

We can now subtract this from the original equation in order to get a linear equation that we can solve:

2x = 4
x = 2

Now that we know the value of x, we can substitute it in the first equation in order to solve it:

4(2) + 2y = 12
2y = 12 - 8
y = \frac{4}{2} = 2

So, the final solution is:

x = 2
y = 2