# How do you solve the following system?: 2x +5y = 3 , x +12y = -12

Nov 16, 2015

$x = \frac{96}{19} \approx 5.05$
$y = - \frac{27}{19} \approx - 1.42$

#### Explanation:

There are different methods to solving a system of equations. Please read it until the end!

Using substitution : Solve for one variable in one equation and then substitute it in the other one, therefore, finding the value of the second variable, and then using it to find the value of the first variable.

$2 x + 5 y = 3$

$x + 12 y = - 12$

Solving for $x$ in the first equation:

$2 x = 3 - 5 y$

$x = \frac{3 - 5 y}{2}$

Substituting on the second one:

$\frac{3 - 5 y}{2} + 12 y = - 12$

$3 - 5 y + 24 y = - 24$

$19 y = - 27$

$y = - \frac{27}{19}$

Using the value found for $y$, we find $x$ on any of the two equations. We will use it on the first one:

$2 x + 5 y = 3$

$2 x + 5 \left(- \frac{27}{19}\right) = 3$

$2 x - \frac{135}{19} = 3$

$2 x = \frac{192}{19}$

$x = \frac{96}{19}$

Elimination method: Combining the two equations to eliminate one of the variables, solve for the other one and then using it to find the value of the one we just eliminated.

$2 x + 5 y = 3$

$x + 12 y = - 12$

If we multiply the second equation by 2 and subtract the first and the second one, we will have:

$\setminus q \quad 2 x + 5 y = 3$
$- 2 x - 24 y = + 24$
—————————————
$\setminus q \quad 0 x - 19 y = 27$

$y = - \frac{27}{19}$

Using the value found for $y$, we find $x$ on any of the two equations. Now we will use it the second one, to prove we can use it on any equation :D.

$x + 12 y = - 12$

$x + 12 \left(- \frac{27}{19}\right) = - 12$

$x - \frac{324}{19} = - 12$

$x = \frac{324}{19} - 12 = \frac{324}{19} - \frac{228}{19} = \frac{96}{19}$