# How do you solve the system of equations 10x - 8y = 5 and 15x - 12y = 1?

Jun 23, 2018

See a solution process below:

#### Explanation:

First, multiply the first equation by $\textcolor{red}{3}$:

$\textcolor{red}{3} \left(10 x - 8 y\right) = \textcolor{red}{3} \times 5$

$\left(\textcolor{red}{3} \times 10 x\right) - \left(\textcolor{red}{3} \times 8 y\right) = 15$

$30 x - 24 y = 15$

Next, multiply the second equation by $\textcolor{red}{2}$:

$\textcolor{red}{2} \left(15 x - 12 y\right) = \textcolor{red}{2} \times 1$

$\left(\textcolor{red}{2} \times 15 x\right) - \left(\textcolor{red}{2} \times 12 y\right) = 2$

$30 x - 24 y = 2$

The left side of both equations are equal: $30 x - 24 y$

However, the right side of both equations are not equal: $15 \cong 2$

This indicates the two lines are parallel lines and not the same line.

Therefore, there are no solutions or, in other words, the solution is the null or empty set: $\left\{\emptyset\right\}$