# The elimination method

**The elimination method**

When solving simultaneous equations, if two equations are true for the same values, it is possible to add our subtract then to come up with a third equation. For example,

3*x* + *y* = 9

5*x* – *y* = 7

Adding both equations together will provide the following solution:

8*x* = 16 (the *x* and the integers are added while, the *y *is eliminated)

*x* = 2

To find the solution of *y* when *x* = 2, this is substituted into one of the equations:

3*x* + *y* = 9

3 x 2 + *y* = 9

6 + *y* = 9

*y* = 3

To check that *x* = 2 and *y* = 3, substitute these values in the other equation:

5*x* – *y* = 7

5 x 2 – *3* = 7

10 – *3* = 7

7 = 7

**The elimination method**

In another scenario, the multiplication of one or both of the equations may be necessary to eliminate one of the variables. For example

4*x* – *y* = 29 (equation a)

3*x* + 2*y* = 19 (equation b)

multiply equation a by 2

8*x* – 2*y* = 58

3*x* + 2*y* = 19

Adding both equations together will provide the following solution:

11*x* = 77 (divide both sides by 11)

*x* = 7

To find the solution of *y* when *x* = 7, this is substituted into one of the equations:

4*x* – *y* = 29

4 x 7 – *y* = 29

28 – *y* = 29

*-y* = 1

To check that *x* = 7 and *y* = -1, substitute these values in the other equation:

3*x* + 2*y* = 19

3 x 7 + 2 x -1 = 19

21 -2 = 19

19 = 19