# How do you solve the following system?: -1/2x+5y=-20 , y=1-3x

Jan 10, 2016

Answer given in extreme detail so that you can see what is happening. With practise you should be able to do this in 4 to 5 lines.

color(blue)(color(white)(...)(x,y) -> (1 19/31 ,3 26/31)

#### Explanation:

Given:
$- \frac{1}{2} x + 5 y = - 20. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left(1\right)$
$y = 1 - 3 x \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left(2\right)$

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

Substitute for y in (1) using (2)

$- \frac{1}{2} x + 5 \left(1 - 3 x\right) = - 20. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left({1}_{a}\right)$

Multiplying out the brackets
$- \frac{1}{2} x - 15 x + 5 = - 20$

$- \frac{31}{2} x + 5 = - 20$

Collecting like terms
$- \frac{31}{2} x = - 25$

Multiply both sides by 2
$- 31 x = - 50$

Multiply both sides by (-1)
$31 x = 50$

Divide both sides by 31
$x = \frac{50}{31.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left(3\right)$

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$\textcolor{b l u e}{\text{To find the value of y}}$

Substitute equation (3) into equation (2) giving

$y = 1 - 3 x \to y = 1 - 3 \left(\frac{50}{31}\right)$

$y = 1 - \frac{150}{31}$

$y = - \frac{119}{31} = - 3 \frac{26}{31}$
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$\textcolor{b l u e}{\text{Putting it all together}}$

The point of intersection of the two graphs/equations is:

$\left(x , y\right) \to \left(1 \frac{19}{31} , 3 \frac{26}{31}\right)$