# How do you solve the system of equations 2x+5y=5 and -6x+7y=-37 using elimination?

Jan 30, 2018

$x = 5 \mathmr{and} y = - 1$

#### Explanation:

To solve a system using elimination, the co-efficients of one of the variable must be the same in both equations. It helps if they have opposite signs. Note that we have $+ 2 x \mathmr{and} - 6 x$

The $L C M$ of $2 \mathmr{and} 6$ is $6$

$\textcolor{w h i t e}{\times \times \times} + 2 x + 5 y = 5 \text{ "" } A$
$\textcolor{w h i t e}{\times \times \times} - 6 x + 7 y = - 37 \text{ } B$

$A \times 3 : \rightarrow \text{ "color(blue)(+6x)+15y =15" " " } C$
$\textcolor{w h i t e}{\times \times \times x . \times} \textcolor{b l u e}{- 6 x} + 7 y = - 37 \text{ } B$

The $x$ terms are now additive inverses. They add to $0$

$C + B : \rightarrow \textcolor{w h i t e}{\times \times \times} 22 y = - 22 \text{ } \leftarrow$ no $x$ term.

$\textcolor{w h i t e}{\times \times \times \times \times \times \times x} y = - 1$

Substitute $- 1$ for $y$ in any equation - let's use $A$

$\textcolor{w h i t e}{\times \times \times} 2 x + 5 \left(- 1\right) = 5 \text{ "" } A$
$\textcolor{w h i t e}{\times \times \times \times} 2 x \text{ } - 5 = 5$
$\textcolor{w h i t e}{\times \times \times \times \times \times x} 2 x = 10$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times} x = 5$

Check the values in another equation: Use $B$

$\textcolor{w h i t e}{\times \times \times} - 6 x + 7 y = - 37 \text{ } B$
$\textcolor{w h i t e}{\times \times \times} - 6 \left(5\right) + 7 \left(- 1\right) = - 37$

$\textcolor{w h i t e}{\times \times \times} - 30 - 7 = - 37$
$\textcolor{w h i t e}{\times \times \times \times x} - 37 = - 37 \text{ } \leftarrow$ the equation works out