# How do you solve the system 5x-10y=15 and 3x-2y=3 by multiplication?

Jun 4, 2018

See explanation.

The system is:

## $\left\{\begin{matrix}5 x - 10 y = 15 \\ 3 x - 2 y = 3\end{matrix}\right.$

First we can see that all terms in the first equation are divisible by $5$. So we can divide both sides bu $5$ to get the lower numbers:

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

Now the coefficients of $y$ are the same in both equations $\left(- 2\right)$, so if we multiply any of the equations by $- 1$ we get the opposite coefficients:

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

Now if we add both sides of both equations we get an equation with one unknown only:

## $x = 0$

Now we can substitute the calculated alue of $x$ into any of the previous equations to calculate $y$:

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

Now we can write the answer:

The system has one solution: