# How do you solve the system of linear equations 2x+3y= 7 and -3x-5y= -13?

Aug 15, 2017

$x = - 4 \mathmr{and} y = 5$

#### Explanation:

These equations are ideal for creating additive inverses to eliminate the $x$-terms. Additive inverses add to $0$

$\textcolor{w h i t e}{\times \times \times . \times} \textcolor{b l u e}{2 x} + 3 y = + 7 \textcolor{w h i t e}{\times \times . \times \times \times x} A$
$\textcolor{w h i t e}{\times \times \times . .} \textcolor{b l u e}{- 3 x} - 5 y = - 13 \textcolor{w h i t e}{\times \times \times \times \times x} B$

$A \times 3 : \textcolor{w h i t e}{\times \times x} \textcolor{b l u e}{6 x} + 9 y = + 21 \textcolor{w h i t e}{\times \times \ldots \times \times x} C$
$B \times 2 : \textcolor{w h i t e}{x . x .} \textcolor{b l u e}{- 6 x} - 10 y = - 26 \textcolor{w h i t e}{\times \times \times \times \times} D$

$C + D : \textcolor{w h i t e}{\times \times \times x} - y = - 5 \text{ } \leftarrow$ the $x$ term is $0$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times} y = 5$

Substitute $y = 5$ into A

$\textcolor{w h i t e}{\times \times . \times} 2 x + 3 \left(5\right) = 7 \textcolor{w h i t e}{\times \times \times x} A$

$\textcolor{w h i t e}{\times \times \times \times . . \times} 2 x = - 8$

$\textcolor{w h i t e}{\times \times \times . \times . . \times} x = - 4$

Substitute for $x \mathmr{and} y$ in B to check:

$- 3 \left(- 4\right) - 5 \left(5\right)$

$12 - 25$

$= - 13$

The equation checks out.