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

Feb 3, 2016

$x = \frac{10}{3}$ and $y = \frac{11}{3}$.

#### Explanation:

We have the following system:
$y = 2 x - 3$
$y = 0.5 x + 2$

Notice that both equations have the same variable isolated. Since the first equaion equals y and the second one too, we conclude that:
$2 x - 3 = 0.5 x + 2$

Now we can procced with placing all letters in one side of the equation and all numbers to the other side.
$2 x - 0.5 x = 2 + 3$
$1.5 x = 5$
$x = \frac{5}{1.5}$

Simplify this fraction by multiplying both parts by 2: $x = \frac{10}{3}$.

Now that we have the value for x, we can replace it in any of the two equations. The result must be the same, no matter which one we choose. For the first equation:
$y = 2 \cdot \left(\frac{10}{3}\right) - \frac{3}{1}$
$y = \frac{20}{3} - \frac{9}{3}$
$y = \frac{11}{3}$

And for the second equation:
$y = 0.5 \cdot \left(\frac{10}{3}\right) + \frac{2}{1}$
$y = \frac{5}{3} + \frac{6}{3}$
$y = \frac{11}{3}$.