What are two-variable linear equations and how do I solve them?

A two-variable linear equation system would be something like:

x+y=240

2x-y=150

There are two methods to solve a system like this: substitution and elimination.

With substitution, one equation is rearranged to find the value of one of the variables and then that variable value is substituted into the equation.

Rearrange the first equation

y=240-x

Substitute the value into the second equation

2x - (240-x) = 150

Solve for x.

2x-240+x=150

3x-240=150

3x=390

x=130

Lastly, plug in the value found into one of the equations, and solve for the other varaible.

130+y=240

y=110

The second method, elimination, requires the variable terms of the equations to cancel out. In the example given, the y terms will cancel out when added together. However, I will instead match the x terms together to cancel out so show the process.

Multiply the first equation by - 2

-2x-2y=-480

Add the two equations together, combining like terms.

-2x-2y=-480

2x-y=150

0-3y=-330

Solve for the variable.

-3y=-330

3y=330

y=110

Lastly, plug the found variable into one of the equations to solve for the other.

2x-110=150

2x=260

x=130

So, the answer to the equation system x+y=240, 2x-y=150 is (130,110).