# What does it mean to solve a system of equations by substitution?

Sep 21, 2014

I think that this is better explained with an example.

Lets say that we have the linear system

$x + y = 17$
$x - y = 3$

Neither one of these equation can be solved on their own because an equation with more than one unknown value cannot be solved.

We have to eliminate one of the variable by substitution.

We have 2 options. First, pick one of equations and then solve for either $x$ or $y$.

Lets use the first equation and solve for $x$.

$x + y = 17$

$x = 17 - y$

Now we see that $x$ and the expression $17 - y$ are the same quantity. We will leverage this information with the second equation by replacing the $x$ with $17 - y$.

$x - y = 3 \to$ Second equation

Make substitution for $x$

$\left(17 - y\right) - y = 3$

Combine like terms

$17 - y - y = 3$

$17 - 2 y = 3$

Subtract 17 from both sides

$- 2 y = - 14$

Divide by sides by $- 2$

$\frac{- 2 y}{- 2} = \frac{- 14}{- 2}$

$y = 7$

Now we can find $x$ by substituting in the $7$ for $y$.

$x + 7 = 17$
$x = 10$

Okay, so now we know that $y = 7$ and $x = 10$.

We check our work by substituting those values into the other equation.

$x - y = 3$
$10 - 7 = 3$
$3 = 3 \to$ We are correct