# 5x+5y=40, x+y=8. What is the solution?

Mar 4, 2018

$\left\{8 - t , t | t \in \mathbb{R}\right\}$

#### Explanation:

We have the following system of equations

{: (5x+5y=40), (x+y=8) :}}

Adding $- 1 \text{/} 5$ of eqn. $1$ to eqn. $2$ transforms the system into echelon form as

{:(5x+5y=40),(0=0):}}

To find solutions, we discard the degenerate equation and solve the other equation.

$5 x + 5 y = 40$

Since $y$ is not a leading variable, we set it equal to an arbitrary variable, say $y = t$ for $t \in \mathbb{R}$.

We now solve for $x$

$5 x + 5 t = 40 \Rightarrow 5 x = 40 - 5 t \Rightarrow x = 8 - t$

So the set of solutions is $\left\{8 - t , t | t \in \mathbb{R}\right\}$