# What are the solution(s) of the system of equations 2x+y=1, x-y=3?

Oct 13, 2015

$\left\{\begin{matrix}x = \frac{4}{3} \\ y = - \frac{5}{3}\end{matrix}\right.$

#### Explanation:

Your system of equations looks like this

$\left\{\begin{matrix}2 x + y = 1 \\ x - y = 3\end{matrix}\right.$

Notice that if you add the left-hand sides and the right-hand sides of the two equations separately, the $y$-term will cancel out. This will allow you to find the value of $x$.

$\left\{\begin{matrix}2 x + y = 1 \\ x - y = 3\end{matrix}\right.$
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$2 x + \textcolor{red}{\cancel{\textcolor{b l a c k}{y}}} + x - \textcolor{red}{\cancel{\textcolor{b l a c k}{y}}} = 1 + 3$

$3 x = 4 \implies x = \textcolor{g r e e n}{\frac{4}{3}}$

Pick one of the two equations and replace $x$ with its determined value to get the value of $y$.

$\frac{4}{3} - y = 3$

$4 - 3 y = 9$

$- 3 y = 5 \implies y = \textcolor{g r e e n}{- \frac{5}{3}}$

Therefore, the solution set for this system of equations is

$\left\{\begin{matrix}x = \frac{4}{3} \\ y = - \frac{5}{3}\end{matrix}\right.$