Let's discuss the different ways of solving a system of linear equations.

Addition/Subtraction

If the coefficients of a variable are equal to each other in both equations, we can add or subtract the equations. For example,

{2 y + 3 x = 8 5 y - 3 x = 1

In this system, the

x

variable has the same coefficient in both equations but with opposite signs,

3 x

and

- 3 x .

Therefore, we can add these two equations. Then we will obtain an equation with one unknown:

7 y = 9 .

This equation gives the value for

y

as

\frac{9}{7}

. Using this solution in one of the original equations, we can find the corresponding

x -

value.

Multiplication

If the coefficients of the same variables are not the same, we should first multiply one or both of them by a number so that we obtain their least common multiple. For example,

{2 x + 3 y = 8 3 x - y = 1

In this system, the variables have different coefficients. Therefore, we need to find the least common multiple of at least one of them. Let's multiply the first equation by

3

and the second by

2 .

{3 \cdot (2 x + 3 y) = 3 \cdot 8 2 \cdot (3 x - y) = 2 \cdot 1 \Rightarrow {6 x + 9 y = 24 6 x - 2 y = 2

Now, since the

x

variable has the same coefficient in both of the equations, we can subtract one from the other. Then we have:

11 y = 22 .

This equation gives the value for

y

as

2 .

Using this solution in one of the original equations, we can find the corresponding

x -

value.

Exercise