# What is the solution for 2x + y = 5 and 2x - 5y = 1?

Feb 25, 2017

See the entire solution process below:

#### Explanation:

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

$2 x + y = 5$

$2 x - \textcolor{red}{2 x} + y = 5 - \textcolor{red}{2 x}$

$0 + y = 5 - 2 x$

$y = 5 - 2 x$

Step 2) Substitute $5 - 2 x$ for $y$ in the second equation and solve for $x$:

$2 x - 5 y = 1$ becomes:

$2 x - 5 \left(5 - 2 x\right) = 1$

$2 x - 25 + 10 x = 1$

$12 x - 25 = 1$

$12 x - 25 + \textcolor{red}{25} = 1 + \textcolor{red}{25}$

$12 x - 0 = 26$

$12 x = 26$

$\frac{12 x}{\textcolor{red}{12}} = \frac{26}{\textcolor{red}{12}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{12}}} x}{\cancel{\textcolor{red}{12}}} = \frac{13}{6}$

$x = \frac{13}{6}$

Step 3) Substitute $\frac{13}{6}$ for $x$ in the solution to the first equation at the end of Step 1 and calculate $y$:

$y = 5 - 2 x$ becomes:

$y = 5 - \left(2 \times \frac{13}{6}\right)$

$y = 5 - \frac{26}{6}$

$y = \left(\frac{6}{6} \times 5\right) - \frac{26}{6}$

$y = \frac{30}{6} - \frac{26}{6}$

$y = \frac{4}{6}$

$y = \frac{2}{3}$

The solution is $x = \frac{13}{6}$ and $y = \frac{2}{3}$ or $\left(\frac{13}{6} , \frac{2}{3}\right)$