# How do you find an ordered pair for 2x-3y=7 and -3x+y=7?

Jan 22, 2017

$\left(- 4 , - 5\right)$

#### Explanation:

You are trying to find a single point that will satisfy both equations. In other words, the solution is the point where these two lines cross.

There are several methods you can use to solve this system of two equations and two variables. In this case, substitution may be easiest:

1. Rewrite the second equation to "solve for $y$":
$- 3 x + y = 7$
$- 3 x \textcolor{red}{+ 3 x} + y = 7 \textcolor{red}{+ 3 x}$
$\textcolor{b l u e}{y = 3 x + 7}$

2. Substitute $\textcolor{b l u e}{3 x + 7}$ for $y$ in the first equation:
$2 x - 3 y = 7$
$2 x - 3 \left(\textcolor{b l u e}{3 x + 7}\right) = 7$

3. Expand and simplify
$2 x - 3 \left(3 x + 7\right) = 7$
$2 x - \textcolor{red}{9 x - 21} = 7$
$\textcolor{red}{- 7 x} - 21 = 7$

4. Solve for $x$
$- 7 x - 21 \textcolor{red}{+ 21} = 7 \textcolor{red}{+ 21}$
$- 7 x = \textcolor{red}{28}$
$\frac{- 7 x}{\textcolor{red}{- 7}} = \frac{28}{\textcolor{red}{- 7}}$
$\frac{\cancel{- 7} x}{\cancel{- 7}} = \textcolor{red}{- 4}$
$\textcolor{b l u e}{x = - 4}$

5. Substitute $\textcolor{b l u e}{x = - 4}$ into one of the equations to find $y$. Let's use the form we found in Step 1
$y = 3 x + 7$
$y = 3 \left(\textcolor{red}{- 4}\right) + 7$
$y = \textcolor{red}{- 12} + 7$
$\textcolor{b l u e}{y = - 5}$

Our solution is $\left(- 4 , - 5\right)$

CHECK
Plug the solution into the original equations to make sure they work

2x-3y=?
$2 \left(\textcolor{red}{- 4}\right) - 3 \left(\textcolor{red}{- 5}\right) =$
$\textcolor{red}{- 8} - \textcolor{red}{- 15} = \textcolor{b l u e}{7}$ ok

-3x+y=?
$- 3 \left(\textcolor{red}{- 4}\right) + \left(\textcolor{red}{- 5}\right) =$
$\textcolor{red}{12} + \left(- 5\right) = \textcolor{b l u e}{7}$ ok