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,
{2y+3x=85y−3x=1
In this system, the
x variable has the same coefficient in both equations but with opposite signs,
3x and
-3x. Therefore, we can add these two equations. Then we will obtain an equation with one unknown:
7y=9.
This equation gives the value for
y as
79. 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,
{2x+3y=83x−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⋅(2x+3y)=3 ⋅ 82⋅(3x−y)=2 ⋅ 1⇒{6x+9y=246x−2y=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:
11y=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.