How do you solve the following system?: 61x -31y =-33 , -3x +7y = 8

Mar 20, 2016

$\left(x , y\right) = \left(\frac{17}{334} , \frac{389}{334}\right)$

Explanation:

We have:

$\left\{\begin{matrix}61 x - 31 y = - 33 \text{ "" "" "" "" "mathbf(eq. 1) \\ -3x+7y=8" "" "" "" "" "" } \textcolor{w h i t e}{s l} m a t h b f \left(e q . 2\right)\end{matrix}\right.$

We want to either cancel out the $x$ terms or the $y$ terms. We can cancel the $x$ terms by multiplying $m a t h b f \left(e q . 1\right)$ by $3$ and $m a t h b f \left(e q . 2\right)$ by $61$, and then adding the two.

$\left.\begin{matrix}\textcolor{w h i t e}{\text{-.-"183x-93y=-99" "" "" "" "color(white)(sl)mathbf(eq. 1)xx3 \\ ul(-183x+427y=488" "+)" "" "color(white)(ss)mathbf(eq. 2)xx61 \\ color(white)("-------------")334y=389" "" "" "" "" }} m a t h b f \left(e q . 3\right)\end{matrix}\right.$

From $m a t h b f \left(e q . 3\right)$, we can solve for $y$ to see that

color(blue)(y=389/334

We can now plug this value of $y$ into either of the original two equations. I will choose $m a t h b f \left(e q . 2\right)$ since the numbers will be smaller.

$- 3 x + 7 \textcolor{b l u e}{y} = 8 \text{ "=>" } - 3 x + 7 \left(\textcolor{b l u e}{\frac{389}{334}}\right) = 8$

Solving this equation, we multiply $7$ and $389 / 334$:

$- 3 x + \frac{2723}{334} = 8$

$- 3 x = 8 - \frac{2723}{334}$

Find a common denominator.

$- 3 x = \frac{2672}{334} - \frac{2723}{334}$

$- 3 x = - \frac{51}{334}$

$x = - \frac{51}{334} \left(- \frac{1}{3}\right)$

The negatives will cancel, so $x$ will be positive, and note that $51 = 3 \times 17$, so the final value of $x$ is:

color(red)(x=17/334

Written as the ordered pair $\left(x , y\right)$, our final answer is

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{{\int}^{\int}} \left(x , y\right) = \left(\frac{17}{334} , \frac{389}{334}\right) \textcolor{w h i t e}{{\int}^{\int}} |}}}$