# How do you solve the following system?:  -x+4y=8 , x=2y+1

Aug 21, 2017

See a solution process below:

#### Explanation:

Step 1) Because the second equation is already solved for $x$ we can substitute $\left(2 y + 1\right)$ for $x$ in the first equation and solve for $y$:

$- x + 4 y = 8$ becomes:

$- \left(2 y + 1\right) + 4 y = 8$

$- 2 y - 1 + 4 y = 8$

$4 y - 2 y - 1 = 8$

$\left(4 - 2\right) y - 1 = 8$

$2 y - 1 = 8$

$2 y - 1 + \textcolor{red}{1} = 8 + \textcolor{red}{1}$

$2 y - 0 = 9$

$2 y = 9$

$\frac{2 y}{\textcolor{red}{2}} = \frac{9}{\textcolor{red}{2}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} y}{\cancel{\textcolor{red}{2}}} = \frac{9}{2}$

$y = \frac{9}{2}$

Step 2) Substitute $\frac{9}{2}$ for $y$ in the second equation and calculate $x$:

$x = 2 y + 1$ becomes:

$x = \left(2 \cdot \frac{9}{2}\right) + 1$

$x = \left(\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \cdot \frac{9}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}}}\right) + 1$

$x = 9 + 1$

$x = 10$

The Solution Is:

$x = 10$ and $y = \frac{9}{2}$ or $\left(10 , \frac{9}{2}\right)$

Aug 21, 2017

$\left(x , y\right) \to \left(10 , \frac{9}{2}\right)$

#### Explanation:

$- \textcolor{red}{x} + 4 y = 8 \to \left(1\right)$

$\textcolor{red}{x} = 2 y + 1 \to \left(2\right)$

$\text{substitute "color(red)(x)=2y+1" into } \left(1\right)$

$\Rightarrow - \left(2 y + 1\right) + 4 y = 8$

$\Rightarrow - 2 y - 1 + 4 y = 8$

$\Rightarrow 2 y = 9 \Rightarrow y = \frac{9}{2}$

$\text{substitute this value into } \left(2\right)$

$\Rightarrow x = \left(2 \times \frac{9}{2}\right) + 1 = 9 + 1 = 10$

$\textcolor{b l u e}{\text{As a check}}$

$\text{substitute these values into } \left(1\right)$

$- 10 + \left(4 \times \frac{9}{2}\right) = - 10 + 18 = 8 \leftarrow \text{ True}$

$\Rightarrow \text{point of intersection } = \left(10 , \frac{9}{2}\right)$

Aug 21, 2017

$x = 10 \mathmr{and} y = 4.5$

#### Explanation:

Notice that the $x$-terms are ADDITIVE INVERSES. That means that they will add together to give $0$.

Change the second equation into the same form so we have:

$\textcolor{w h i t e}{\times \times \times} \textcolor{b l u e}{- x} + 4 y = 8 \text{ } \ldots \ldots . A$
$\textcolor{w h i t e}{\times \times \times} \textcolor{b l u e}{+ x} - 2 y = 1 \text{ } \ldots \ldots \ldots B$

$A + B : \text{ } \textcolor{b l u e}{0 x} + 2 y = 9$

$\textcolor{w h i t e}{\times \times \times \times \times x} y = 4.5$

Substitute this value for $y$ into the original equation for $x$

$x = 2 y + 1$

$x = 2 \left(4.5\right) + 1$

$x = 10$

Check in the other equation:$\text{ } - x + 4 y = 8$

$- \left(10\right) + 4 \left(4.5\right)$

$- 10 + 18$

$= 8 \text{ } \leftarrow$ the answer is correct.