# How do you solve the following system:  3x+5y=5 , 4x+y=10 ?

Jan 23, 2017

See the entire solution process below:

#### Explanation:

Step 1) Solve the second equation for $y$:

$4 x + y - \textcolor{red}{4 x} = 10 - \textcolor{red}{4 x}$

$4 x - \textcolor{red}{4 x} + y = 10 - \textcolor{red}{4 x}$

$0 + y = 10 - 4 x$

$y = 10 - 4 x$

Step 2) Substitute $\textcolor{red}{10 - 4 x}$ for $y$ in the first equation and solve for $x$:

$3 x + 5 \left(10 - 4 x\right) = 5$

$3 x + 50 - 20 x = 5$

$3 x - 20 x + 50 = 5$

$- 17 x + 50 = 5$

$- 17 x + 50 + \textcolor{red}{17 x} - \textcolor{b l u e}{5} = 5 + \textcolor{red}{17 x} - \textcolor{b l u e}{5}$

$- 17 x + \textcolor{red}{17 x} + 50 - \textcolor{b l u e}{5} = 5 - \textcolor{b l u e}{5} + \textcolor{red}{17 x}$

$0 + 50 - 5 = 0 + 17 x$

$45 = 17 x$

$\frac{45}{\textcolor{red}{17}} = \frac{17 x}{\textcolor{red}{17}}$

$\frac{45}{17} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{17}}} x}{\cancel{\textcolor{red}{17}}}$

$\frac{45}{17} = x$

$x = \frac{45}{17}$

Step 3) Substitute $\textcolor{red}{\frac{45}{17}}$ for $x$ in the solution to the second equation at the end of Step 1:

$y = 10 - \left(4 \times \frac{45}{17}\right)$

$y = \left(\frac{17}{17} \times 10\right) - \left(\frac{180}{17}\right)$

$y = \frac{170}{17} - \frac{180}{17}$

$y = - \frac{10}{17}$

The solution is: $x = \frac{45}{17}$, $y = - \frac{10}{17}$ or $\left(\frac{45}{17} , - \frac{10}{17}\right)$