# How do you solve the following system?: 2x -5y =9 , 5x -3y = 2

Nov 18, 2015

$y = - \frac{41}{19} \textcolor{w h i t e}{\times \times} x = - \frac{17}{19}$

Keeping in fraction form improves precision

#### Explanation:

Given:
$2 x - 5 y = 9. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left(1\right)$
$5 x - 3 y = 2. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left(2\right)$

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$\textcolor{b l u e}{\text{Consider Equation (1)}}$

Add $5 y$ to both sides
$2 x = 9 + 5 y$

Subtract 9 from both sides.
$2 x - 9 = 5 y$

Divide both sides by 5
$\frac{2}{5} x - \frac{9}{5} = y$

Write in conventional form
$\textcolor{g r e e n}{y = \frac{2}{5} x - \frac{9}{5.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left({1}_{a}\right)}$
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$\textcolor{b l u e}{\text{Consider Equation (2)}}$

Add $3 y$ to both sides
$5 x = 2 + 3 y$

Subtract 2 from both sides
$5 x - 2 = 3 y$

Divide both sides by 3
$\frac{5}{3} x - \frac{2}{3} = y$

Write in conventional form
$\textcolor{g r e e n}{y = \frac{5}{3} x - \frac{2}{3.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left({2}_{a}\right)}$
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To end up with just 1 variable thus solvable.

$\textcolor{g r e e n}{\text{Equation "(2_a))color(red)( =y= )color(blue)("Equation "(1_a)}}$

$\textcolor{g r e e n}{\frac{2}{5} x - \frac{9}{5}} \textcolor{red}{=} \textcolor{b l u e}{\frac{5}{3} x - \frac{2}{3}}$

Collecting like terms
$\frac{2}{5} x - \frac{5}{3} x = \frac{9}{5} - \frac{2}{3}$

$\frac{6 - 25}{15} x = \frac{27 - 10}{15}$

$- \frac{19}{15} x = \frac{17}{15}$

$- 19 x = 17$

$\textcolor{red}{x = - \frac{17}{19.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left(3\right)}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Substitute for $x$ in Equation (1)

$2 x - 5 y = 9. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left(1\right)$

$2 \textcolor{red}{\left(- \frac{17}{19}\right)} - 5 y = 9$

$5 y = 2 \left(- \frac{17}{19}\right) - 9$

$y = \frac{2}{5} \left(- \frac{17}{19}\right) - \frac{9}{5}$

$\textcolor{red}{y = - 2 \frac{3}{19} = - \frac{41}{19.} \ldots \ldots \ldots \ldots . . \left(4\right)}$