This section is rated ADVANCED

 

If you want to find out the value of an unknown, you need to know the value of everything else in the equation. So if you have an equation with two unknowns, like:

 

4x + 12y = 44

 

it is impossible to find unique solutions for x and y. However, if you have a second equation containing the same unknowns, you can use the same techniques as you have been using on single equations to find the values of both unknowns.

 

4x + 12y = 44

7x + 4y = 26

 

The technique used to find the values of the unknowns in these equations is quite simple. You need to change one or both of them so that the multiplier for one of the unknowns is the same in both equations. In this case it is easy to see that the multiplier for y in the second equation is exactly one-third of the value in the first. You know that you can keep the equation balanced if the same operation is performed on both sides. Here, we will multiply both sides by three:

 

3 × (7x + 4y) = 3 × 26

21x + 12y = 78

 

If we subtract one equation from the other, the y terms will cancel out, leaving only x terms. This will leave a simple equation in x.

 

21x + 12y = 78

4x + 12y = 44

 

subtraction leaves:

 

17x = 34

x = 2

 

The solution to x can then be added to either of the original equations to find y:

 

7x + 4y = 26

14 + 4y = 26

y = 3