# How do you solve the following system?: 2x-13y= 9 , 17 x + 2y = 6

Nov 12, 2017

$\left(\frac{32}{75} , - \frac{47}{75}\right)$

#### Explanation:

In solving systems of linear equations, you can use the elimination method, substitution method or graphing. Using any of these three methods would yield the same answer so it is up to you to choose which method you are most comfortable with. However in this case, I would show the elimination method.

*graphing will not be done since it is difficult to show graphs using this medium and since it is difficult to determine the exact solutions if the values are not whole numbers.

So in this case, you are given:
$2 x - 13 y = 9$
$17 x + 2 y = 6$

Elimination Method
$2 x - 13 y = 9$
$17 x + 2 y = 6$

You can either multiply the first equation by 17 and the second by 2 in order to eliminate $x$ or you can multiply 2 to the first equation and 13 to the second to eliminate $y$. For this I will choose the second method since the numbers would be smaller.

$2 \left(2 x - 13 y\right) = 2 \left(9\right)$
$13 \left(17 x + 2 y\right) = 13 \left(6\right)$

Therefore
$4 x - 26 y = 18$
$221 x + 26 y = 78$

Adding the two equations to eliminate $y$ would lead you to this.
$225 x = 96$
$x = \frac{96}{225}$
$x = \frac{32}{75}$

Using the value of $x$ to obtain $y$...
$2 \left(\frac{32}{75}\right) - 13 y = 9$
$\frac{64}{75} - 13 y = 9$
$\frac{64}{75} - 9 = 13 y$
$- \frac{611}{75} = 13 y$
$13 y = - \frac{611}{75}$
$y = - \frac{611}{75} \cdot \frac{1}{13}$
$y = - \frac{47}{75}$

$\left(\frac{32}{75} , - \frac{47}{75}\right)$

Nov 12, 2017

$x = \frac{32}{75}$ and $y = - \frac{47}{75}$

#### Explanation:

You have two equations:

A: $2 x - 13 y = 9$
B: $17 x + 2 y = 6$

Use equation A to find the value of $x$ with respect to $y$:

A: $2 x - 13 y = 9 \rightarrow 2 x = 9 + 13 y \rightarrow x = \frac{9 + 13 y}{2}$

Then use this value in equation B to replace $x$:

B: $17 x + 2 y = 17 \cdot \frac{9 + 13 y}{2} + \frac{4 y}{2} = \frac{153 + 225 y}{2} = 6$
$\rightarrow y = \frac{12 - 153}{225} = - \frac{141}{225} = - \frac{47}{75}$

Finaly, use this value of $y$ to find $x$:

$x = \frac{9 + 13 y}{2} = \frac{9 + 13 \cdot \left(- \frac{47}{75}\right)}{2} = \frac{9 - \frac{611}{75}}{2} = \frac{\frac{64}{75}}{2} = \frac{32}{75}$