Question

How do you solve `y=x-2` and `y=4x+1`?

Answer 1

This is called system of equations because you are searching a pair of numbers that inserted as `x` and `y` verify both equations at the same time.

To solve a system of equation you have to use one equation to express one variable as a function of the other, and substitute the result in the second equation.

In your case you have

`y=x-2`

So your `y` variable will be always equal to `x-2`, even in the second equation. Then we can write

`x-2=4x+1`

Here I replaced the "value" of `y` in the second equation.

Now you have an equation in one variable that you can solve as usual

`x-2=4x+1`
`4x-x=-2-1`
`3x=-3`
`x=-1`

With a value for `x` you can substitute in one of the two equations and obtain `y`. Because both equations has to be valid at the same time, both equations must give you the same number for `y`. Let's try:

`y=x-2`
`y=-1-2`
`y=-3`

and, with the second equation

`y=4x+1`
`y=4(-1)+1`
`y=-4+1`
`y=-3`.

It works! Then our solution is given by `x=-1` and `y=-3`.