# Question #95527

Jan 4, 2018

$\left\{\begin{matrix}x = - 3 \\ y = - 4\end{matrix}\right.$

#### Explanation:

Start by dividing both sides of the second equation by $- 2$. You will end up with

$\frac{2 y}{- 2} = \frac{7}{-} 2 + \frac{5 x}{-} 2$

$- y = - \frac{7}{2} - \frac{5 x}{2}$

Now, notice that the left-hand side of the two equations is equal.

$\left\{\begin{matrix}- y = - \frac{5}{7} - \frac{11 x}{7} \\ - y = - \frac{7}{2} - \frac{5 x}{2}\end{matrix}\right.$

This means that you can equate the two right-hand sides to get an equation with one unknown, $x$.

$- \frac{5}{7} - \frac{11 x}{7} = - \frac{7}{2} - \frac{5 x}{2}$

DIvide or multiply both sides by $- 1$ to get rid of the minus signs.

$\frac{5}{7} + \frac{11 x}{7} = \frac{7}{2} + \frac{5 x}{2}$

$\frac{5 + 11 x}{7} = \frac{7 + 5 x}{2}$

Next, cross multiply, i.e. multiply the numerator of one fraction by the denominator of the second fraction and vice versa. You will end up with

$7 \cdot \left(7 + 5 x\right) = 2 \cdot \left(5 + 11 x\right)$

This will be equivalent o

$49 + 35 x = 10 + 22 x$

Subtract $22 x$ and $49$ from both sides to get

$13 x = - 39$

Finally, divide both sides by $13$ to isolate $x$

$x = - \frac{39}{13} = - 3$

Now that you have a value for $x$, pick one of the two equations and use it to find the value of $y$.

$2 y = 7 + 5 \cdot \left(- 3\right)$

$2 y = - 8$

Divide both sides by $2$ to isolate $y$

$y = - 4$

Therefore, your system of two equations with two unknowns has the solution set

$\left\{\begin{matrix}x = - 3 \\ y = - 4\end{matrix}\right.$

If you want, you can double-check the result by plugging in the two values into the original equations.

You will have

$- \left(- 4\right) = - \frac{5}{7} - \frac{11 \cdot \left(- 3\right)}{7}$

$4 = \frac{- 5 + 33}{7} \text{ } \textcolor{g r e e n}{\sqrt{}}$

and

$2 \cdot \left(- 4\right) = 7 + 5 \cdot \left(- 3\right)$

$- 8 = 7 - 15 \text{ } \textcolor{g r e e n}{\sqrt{}}$