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

Jun 3, 2016

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.

$y = x - 2$

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

$x - 2 = 4 x + 1$

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

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

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

$y = 4 x + 1$
$y = 4 \left(- 1\right) + 1$
$y = - 4 + 1$
$y = - 3$.

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