# How do solve the following linear system?:  4y-16=x , 2x+7=-5y ?

Jun 15, 2017

$x = - 8.31$
$y = \frac{25}{13}$

#### Explanation:

The first equation already has $x$ as the subject. Therefore you substitute that in place of $x$ in the second equation.

$2 \left(4 y - 16\right) + 7 = - 5 y \to 8 y - 32 + 7 = - 5 y$

Rearrange the equation to have all unknowns on one side.

$13 y = 25$

$y = \frac{25}{13}$

Now substitute the $y$ value into the first equation to get $x$.

$4 \left(\frac{25}{13}\right) - 16 = x$

$x = - 8.31$

Jun 15, 2017

(x,y)=color(red)(""(-108/13,25/13))

#### Explanation:

Given
[1]$\textcolor{w h i t e}{\text{XXX}} 4 y - 16 = x$
[2]$\textcolor{w h i t e}{\text{XXX}} 2 x + 7 = - 5 y$

[1]$\rightarrow$
[3]$\textcolor{w h i t e}{\text{XXX}} x = 4 y - 16$

Using [3] we can replace $x$ in [2] with $4 y - 16$
[4]$\textcolor{w h i t e}{\text{XXX}} 2 \left(4 y - 16\right) + 7 = - 5 y$

Expanding and simplifying the left side of [4]
[5]$\textcolor{w h i t e}{\text{XXX}} 8 y - 25 = - 5 y$

Adding $\left(5 y + 25\right)$ to both sides of [5]
[6]$\textcolor{w h i t e}{\text{XXX}} 13 y = 25$

Dividing both sides of [6] by $13$
[7]$\textcolor{w h i t e}{\text{XXX}} y = \frac{25}{13}$

Using [7] we can replace $y$ in [1] with $\frac{25}{13}$
[8]$\textcolor{w h i t e}{\text{XXX}} 4 \cdot \frac{25}{13} - 16 = x$

Simplifying [8]
[9]$\textcolor{w h i t e}{\text{XXX}} x = - \frac{108}{13}$

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The ugliest part is probably verifying this result by putting the derived values for $x$ and $y$ back into the original equations to check that the results are correct.

...with values like these, I would suggest you go to the effort to verify the results (despite the ugliness). Personally I used a spread sheet for this but you might want to use some other method.

Jun 15, 2017

color(red)(y=25/13 or color(red)(1 12/13,color(red)(x=-108/13 orcolor(red)(-8 4/13

#### Explanation:

$4 y - 16 = x$----$\textcolor{red}{\left(1\right)}$

$2 x + 7 = - 5 y$----color(red)((2)

$4 y - x = 16$----color(red)((3)

$- 5 y - 2 x = 7$----$\left(\textcolor{red}{4}\right)$

color(red)((3) xx 5

$20 y - 5 x = 80$----color(red)((5)

color(red)((4) xx 4

$- 20 y - 8 x = 28$----color(red)((6)

color(red)((5)+(6)

$- 13 x = 108$

color(red)(x=-108/13 orcolor(red)(-8 4/13

Substitute color(red)(x=-108/13 in (2)

$2 \left(\textcolor{red}{- \frac{108}{13}}\right) + 7 = - 5 y$

$- \frac{216}{13} + 7 = - 5 y$

Multiply both sides by 13

$- 65 y = - 216 + 91$

$- 65 y = - 125$

$y = {\cancel{125}}^{25} / {\cancel{65}}^{13}$

color(red)(y=25/13 or color(red)(1 12/13

Check:

Substitute color(red)(y=25/13 and color(red)(x=-108/13 in (1)

4(color(red)(25/13))-16=color(red)(-108/13

$\frac{100}{13} - 16 = - \frac{108}{13}$

$\frac{100 - 208 = - 108}{13}$

Multiply both sides by 65

$100 - 208 = - 108$

color(red)(-108=-108