# How do you solve the following system?: -2x -5y =17, 3x -y = -61

Apr 8, 2016

The joint values that satisfy both equations are:

$\left(x , y\right) \to \left(- \frac{322}{17} , \frac{71}{17}\right)$

#### Explanation:

These are equations of straight line graphs. The gradients are different which means that at some point they will cross.

At that instance they will both share the same values for $x$ and $y$. It is what you are determining when solving such a system

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Given

$- 2 x - 5 y = 17 \text{ }$.................................(1)
$3 x - y = - 61 \text{ }$.................................(2)

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$\textcolor{b l u e}{\text{Determine the value of } x}$

Write equation (2) as :

$y = 3 x + 61 \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left({2}_{a}\right)$

Using $\left({2}_{a}\right)$ substitute for $y$ in equation (1)
so that we have only 1 unknown variable

$- 2 x - 5 \left(3 x + 61\right) = 17$

Multiply out the brackets

$- 2 x - 15 x - 305 = 17$

$- 17 x = 322$

Multiply both sides by (-1)

$+ 17 x = - 322$

$\textcolor{b l u e}{x = - \frac{322}{17}}$
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$\textcolor{b l u e}{\text{Determine the value of } y}$

Substitute for $x$ in equation (1)

$- 2 \left(- \frac{322}{17}\right) - 5 y = 17$

$\frac{644}{17} - 17 = 5 y$

$\textcolor{b l u e}{y = \frac{355}{17} \times \frac{1}{5} = \frac{71}{17}}$ 