How do you solve 2x + y = 3, x - 3y = 5?

May 18, 2018

See a solution process below:

Explanation:

Step 1) Solve the first equation for $y$:

$2 x + y = 3$

$2 x - \textcolor{red}{2 x} + y = 3 - \textcolor{red}{2 x}$

$0 + y = 3 - 2 x$

$y = 3 - 2 x$

Step 2) Substitute $\left(3 - 2 x\right)$ for $y$ in the second equation and solve for $x$:

$x - 3 y = 5$ becomes:

$x - 3 \left(3 - 2 x\right) = 5$

$x - \left(3 \cdot 3\right) + \left(3 \cdot 2 x\right) = 5$

$x - 9 + 6 x = 5$

$1 x + 6 x - 9 = 5$

$\left(1 + 6\right) x - 9 = 5$

$7 x - 9 = 5$

$7 x - 9 + \textcolor{red}{9} = 5 + \textcolor{red}{9}$

$7 x - 0 = 14$

$7 x = 14$

$\frac{7 x}{\textcolor{red}{7}} = \frac{14}{\textcolor{red}{7}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{7}}} x}{\cancel{\textcolor{red}{7}}} = 2$

$x = 2$

Step 3) Substitute $2$ for $x$ in the solution to the first equation at the end of Step 1 and calculate $y$:

$y = 3 - 2 x$ becomes:

$y = 3 - \left(2 \cdot 2\right)$

$y = 3 - 4$

$y = - 1$

The Solution Is:

$x = 2$ and $y = - 1$

Or

$\left(2 , - 1\right)$