# How do you solve using the elimination method 3x + 4y = 5, 6x + 8y = 10?

Sep 1, 2015

This system of equations has an infinite number of solutions.

#### Explanation:

Your system of equations looks like this

$\left\{\begin{matrix}3 x + 4 y = 5 \\ 6 x + 8 y = 10\end{matrix}\right.$

Notice that if you multiply the first equation by $\left(- 2\right)$ and then add the left-hand sides and the right-hand sides separately, you will get

$\left\{\begin{matrix}3 x + 4 y = 5 | \cdot \left(- 2\right) \\ 6 x + 8 y = 10\end{matrix}\right.$

$\left\{\begin{matrix}- 6 x - 8 y = - 10 \\ 6 x \text{ "+ 8y = " } 10\end{matrix}\right.$
stackrel("---------------------------------------------")
$\text{ "0 " "+ 0 " } = 0$

Since you end up with $0 = 0$, which is an equality that does not depends on any variable, the system of equations will have an infinite number of solutions.

You can think of a system of equations as describing two lines. In your case, both equations decribe the same line, because if you multiply the first equation by $2$ you get the second equation

$\left\{\begin{matrix}6 x + 8 y = 10 \\ 6 x + 8 y = 10\end{matrix}\right.$

This implies that you have an infinite number of $\left(x , y\right)$ points that correspond to this line.