# How do you solve the system 4x-5y=-7 and y=5x using substitution?

Oct 1, 2016

$x = \frac{1}{3} \mathmr{and} y = \frac{5}{3}$

#### Explanation:

There are two variables, therefore we need two equations.

$4 x - 5 y = - 7$ and $\textcolor{red}{y = 5 x}$

$\textcolor{red}{y = 5 x}$ means that $y \mathmr{and} 5 x$ have exactly the same value.

$\textcolor{red}{y \text{ and } 5 x}$ are therefore interchangeable.

$4 x - 5 \textcolor{red}{y} = - 7$ has two variables and therefore cannot be solved with one unique solution. There are infinitely many answers.

Replace $\textcolor{red}{y}$ by $\textcolor{red}{5 x}$
This gives an equation with only ONE variable and it can be solved.

$\text{ } 4 x - 5 \textcolor{red}{y} = - 7$
$\textcolor{w h i t e}{\times \times \times x} \downarrow$
$\rightarrow 4 x - 5 \textcolor{red}{\left(5 x\right)} = - 7 \text{ }$ (now the equation has only x)

$4 x - 25 x = - 7$

$- 21 x = - 7 \text{ } \leftarrow \div - 21$

$x = \frac{1}{3}$

Now find the value of $y$, knowing that $y = 5 x$

$y = 5 \left(\frac{1}{3}\right)$

$y = \frac{5}{3}$