# How do you solve the system of equations -6x + 5y = 8 and x + 5y = - 13?

Dec 19, 2017

$x = - 3 \mathmr{and} y = - 2$

#### Explanation:

Let,
$- 6 x + 5 y = 8$ ----- (1)

$x + 5 y = - 13$----(2)

(2)$\implies x = - 13 - 5 y$

Substitute this value of $x$ in (1):

(1)$\implies - 6 \left(- 13 - 5 y\right) + 5 y = 8$

$\implies 78 + 30 y + 5 y = 8$

$\implies 35 y = 8 - 78$

$\implies y = - \frac{70}{35} = - 2$

$\therefore y = - 2$

Now substitute this value in the expression of $x$:

$\implies x = - 13 - 5 y$

$\implies x = - 13 - 5 \left(- 2\right)$

$\implies x = - 13 + 10$

$\therefore x = - 3$

Answer: $x = - 3 \mathmr{and} y = - 2$

Dec 19, 2017

$x = - 3 \mathmr{and} y = - 2$

#### Explanation:

Notice that the $y$ terms both have the same coefficient.

Subtracting the equations will therefore eliminate the $y$ terms.

$\text{ "x" "color(red)(+" "5y)" " =-13" } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . A$
$- 6 x \text{ "color(red)(+" "5y)" " =8" } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . B$

Subtract $A - B \text{ }$ (change the signs of $B$)

$\text{ "x" "+" "5y" " =-13" } \textcolor{w h i t e}{\times \times \times \times x} A$
ul(color(blue)(+)6x" "color(blue)(-)" "5y" " =color(blue)(-)8)" "color(white)(xxxxxxxxx)B
" "7xcolor(white)(xxxxxxxxx) =-21" "color(white)(xxxxxxxxxx)C" "larr div 7
$\text{ } x \textcolor{w h i t e}{\times \times \times \times x} = - 3$

Substitute $- 3 \text{ for } x$ in $A$

$- 3 + 5 y = - 13$
$\text{ } 5 y = - 13 + 3$
$\text{ } 5 y = - 10$
$\text{ } y = - 2$

An alternative method is to equate the $y$ terms.
Transpose the equations to isolate $5 y$ in each:

$5 y = - 13 - x \text{ and } 5 y = 8 + 6 x$

We know that $\text{ } 5 y = 5 y$

Therefore: $\text{ "6x+8 = -13-x" } \leftarrow$ now solve for $x$

$6 x + x = - 13 - 8$
$\text{ } 7 x = - 21$
$\text{ } x = - 3$

Then proceed as described above to get $y = - 2$

Check in $B$
$- 6 x + 5 y$
$= - 6 \left(- 3\right) + 5 \left(- 2\right)$
$= 18 - 10$
$= 8 \text{ } \leftarrow$ the answer is the same as the RHS