How do solve the following linear system?:  y=-6x-9 , -9x+5y=1 ?

Nov 7, 2016

I found:
$x = - \frac{46}{39}$
$y = - \frac{25}{13}$

Explanation:

You can try by taking the value for $y$ from the first equation and substitute into the second as:
$- 9 x + 5 \left(\textcolor{red}{- 6 x - 9}\right) = 1$
rearrange and solve for $x$:
$- 9 x - 30 x - 45 = 1$
$- 39 x = 46$
$x = - \frac{46}{39}$
we use this value for $x$ back into the first equation to find $y$:
$y = - 6 \left(\textcolor{b l u e}{- \frac{46}{39}}\right) - 9 = \frac{276 - 351}{39} = - \frac{75}{39} = - \frac{25}{13}$

Nov 7, 2016

$\left(- \frac{46}{39} , - \frac{25}{13}\right)$

Explanation:

Given the following linear system, you may choose from either performing the substitution method or the elimination method. For this example I would show both and it would be up to you to choose which method you are more comfortable with. However note that using any of the two aforementioned method would produce the same result.

$y = - 6 x - 9$
$- 9 x + 5 y = 1$

Substitution Method:
For this method, since the $y$ in the first equation is already isolated, we may use this value to substitute the value of $y$ in the second equation. In this case, the second equation would be:

$- 9 x + 5 \left(- 6 x - 9\right) = 1$
$- 9 x + \left(- 30 x - 45\right) = 1$
$- 9 x - 30 x - 45 = 1$
$- 39 x = 1 + 45$
$- 39 x = 46$
$x = - \frac{46}{39}$

Since we now have the value of $x$ we can substitute this to the first equation to get the $y$ value.

$y = - 6 \left(- \frac{46}{39}\right) - 9$
$y = \frac{92}{13} - \frac{117}{13}$
$y = - \frac{25}{13}$

In the same way, when we use the Elimination Method, we would arrive at the same answer.

Elimination Method
$- 6 x - y = 9$
$- 9 x + 5 y = 1$

Multiplying the first equation by 5
$- 30 x - 5 y = 45$
$- 9 x + 5 y = 1$

Adding the two equations to eliminate $y$
$- 39 x = 46$
$x = - \frac{46}{39}$

Computing for $y$

$y = - 6 \left(- \frac{46}{39}\right) - 9$
$y = \frac{92}{13} - \frac{117}{13}$
$y = - \frac{25}{13}$

Hence the ordered pair/solution to this linear equation is:
$\left(- \frac{46}{39} , - \frac{25}{13}\right)$