# How do you determine the solution in terms of a system of linear equations for 5x + 2y = 5, 3x + y = 2?

Sep 23, 2015

$x = - 1$
$y = 5$

#### Explanation:

You can solve for the solution using elimination or substitution. Personally, I think it will be easier to use substitution for this problem, so that's what I will use.

First, let's look at the second equation. You will notice that it is easy to isolate $y$ here.
$3 x + y = 2$
$3 x + y - 3 x = 2 - 3 x$
$\textcolor{red}{y = 2 - 3 x}$

Now that we've isolated $y$ in the second equation, we will now substitute this into $y$ in the first equation.
$5 x + 2 \textcolor{red}{y} = 5$
$5 x + 2 \textcolor{red}{\left(2 - 3 x\right)} = 5$
$5 x + 4 - 6 x = 5$
$4 - x + x = 5 + x$
$4 = 5 + x$
$4 - 5 = 5 + x - 5$
$- 1 = x$
$\textcolor{b l u e}{x = - 1}$

Now that we have solved for the value of $x$, let's go back to the equation earlier where we isolated $y$.
$\textcolor{red}{y = 2 - 3} \textcolor{b l u e}{x}$
$y = 2 - 3 \textcolor{b l u e}{\left(- 1\right)}$
$y = 2 + 3$
$\textcolor{red}{y = 5}$

So the solutions are: $\textcolor{b l u e}{x = - 1}$ and $\textcolor{red}{y = 5}$.

Checking
In case you are unsure of your answer, you can check this by substituting the solutions in both of the equations.
$5 \textcolor{b l u e}{x} + 2 \textcolor{red}{y} = 5$
$5 \textcolor{b l u e}{\left(- 1\right)} + 2 \textcolor{red}{\left(5\right)} = 5$
$- 5 + 10 = 5$
$\textcolor{m a \ge n t a}{5 = 5}$

$3 \textcolor{b l u e}{x} + \textcolor{red}{y} = 2$
$3 \textcolor{b l u e}{\left(- 1\right)} + \textcolor{red}{\left(5\right)} = 2$
$- 3 + 5 = 2$
$\textcolor{m a \ge n t a}{2 = 2}$