# How do you solve the following linear system: 3x + 5y = -1, 5x - y = 3?

Jan 12, 2016

$\left(x , y\right) = \left(\frac{1}{2} , - \frac{1}{2}\right)$

#### Explanation:

Given
[1]$\textcolor{w h i t e}{\text{XXX}} 3 x + 5 y = - 1$
[2]$\textcolor{w h i t e}{\text{XXX}} 5 x - y = 3$

By substitution
Re-write [2] as
[3]$\textcolor{w h i t e}{\text{XXX}} y = 5 x - 3$

Substitute $\left(5 x - 3\right)$ for $y$ in [1]
[4]$\textcolor{w h i t e}{\text{XXX}} 3 x + 5 \left(5 x - 3\right) = - 1$

Simplify
[5]$\textcolor{w h i t e}{\text{XXX}} 3 x + 25 x - 15 = - 1$

[6]$\textcolor{w h i t e}{\text{XXX}} 28 x = 14$

[7]$\textcolor{w h i t e}{\text{XXX}} x = \frac{1}{2}$

Substitute $\frac{1}{2}$ for $x$ in [2]
[8]$\textcolor{w h i t e}{\text{XXX}} 5 \left(\frac{1}{2}\right) - y = 3$

[9]$\textcolor{w h i t e}{\text{XXX}} y = \frac{5}{2} - 3 = - \frac{1}{2}$

Jan 12, 2016

$\left(x , y\right) \to \left(\frac{1}{2} , - \frac{1}{2}\right)$
$\textcolor{b r o w n}{\text{Finding x is explained in detail. Finding y is explained using shortcuts.}}$

#### Explanation:

Given:

$\textcolor{b r o w n}{3 x + 5 y = - 1}$ ........................................(1)

$\textcolor{b r o w n}{5 x - y = 3}$..................................................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{To find } x}$

From (2) we have $y = 5 x - 3. \ldots \ldots \ldots . \left({2}_{a}\right)$

Substitute $\left({2}_{a}\right)$into (1) giving:

$3 x + 5 \left(5 x - 3\right) = - 1$

$3 x + 25 x - 15 = - 1$

$\textcolor{b r o w n}{28 x - 15 = - 1}$
'-----------------------------------------------------------
Add $\textcolor{b l u e}{15}$ to both sides giving:

$\textcolor{b r o w n}{28 x - 15 \textcolor{b l u e}{+ 15} = - 1 \textcolor{b l u e}{+ 15}}$

$28 x = 14$
'--------------------------------------------------------
Divide both sides by 28 which is the same as $\textcolor{b l u e}{\times \frac{1}{28}}$

$\frac{28}{28} \times x = 14 \times \frac{1}{28}$

but $\frac{28}{28} = 1 \text{ and } \frac{14}{28} = \frac{1}{2}$ giving:

$\textcolor{m a \ge n t a}{x = \frac{1}{2}}$...........................................(3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{To find y}}$ (Using shortcuts)

Substitute (3) into (1) or (2): Using (2) is easier!

$5 x - y = 3 \to 5 \left(\frac{1}{2}\right) - y = 3$
$\textcolor{m a \ge n t a}{y = 2 \frac{1}{2} - 3 = - \frac{1}{2}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Putting it all together}}$

$\left(x , y\right) \to \left(\frac{1}{2} , - \frac{1}{2}\right)$