Simultaneous Equations Part B: Elimination Method

Hi again, last time we talked about solving linear equations simultaneously by the method of substitution. Now I am going to teach you the other method: elimination. As I said before any method can be used to solve any simultaneous equation so I am going to use the same question to demonstrate the “elimination method”. The trick to it is knowing which method is most suitable to use, usually it is the elimination method, but the safer method to use is the “substitution method”. This is because the “substitution method” is less liable to encounter errors. Let’s look at the same question as before:


y = -x + 8…………………eq’n 1
4y = 3x – 24………………eq’n 2

Elimination means getting rid of one term, either the x or y. The elimination process must occur only when the same term exist in both equations. What do I mean by this?? Let’s take a look. Let’s eliminate the y-term.

First step: multiply equation y = -x + 8 by 4. By doing this all terms in this equation must increase by a factor of 4. The equation now becomes 4y = -4x + 32…..let’s call it equation 3. Note that the equation has not been altered by its numerical value.

Second step: Minus equation 2 from equation 3. This is where a lot of people encounter difficulties, but those with difficulties at this point, I will teach you a nice technique to solve your problems. All you have to do is change all the signs of all the terms of the subtractor.

4y = -4x + 32……………..eq’n 3
(-) = (-) + (-)
4y = 3x – 24………………eq’n 2

4y - 4y = 0, - 4x - 3x = -7x and 32 – (-24)= 32 + 24 = 56…….by subtracting both equations I eliminated the y-term.
4y = -4x + 32……………..eq’n 3
(-) = (-) + (-)
4y = 3x – 24………………eq’n 2
0 = -7x +56
7x = 56
x = 8

Now substitute x = 8 into the equation y = -x + 8 which results to y = 0.

What if you were given a pair of simultaneous equations like the one below:

2y = 3x +7………….eq’n 1
5y = 15x + 10………eq’n 2

To do elimination we must get the same term in both equations. So multiply equation 1 by a factor of 5 and multiply equation 2 by a factor of 2.

2y = 3x +7………….eq’n 1……..x 5
5y = 15x + 10………eq’n 2……..x 2

10y = 15x + 35……..eq’n 3
10y = 30x + 20……..eq’n 4

Now we have the same y-term in both equations now repeat the same process as above.