# How do you solve by substitution 1/2x+2y=12 and x-2y=6?

Jul 10, 2018

$x = 12 , y = 3$

#### Explanation:

Plugging the second equation

$x = 6 + 2 y$ in the first one we get

$\frac{1}{2} \cdot \left(6 + 2 y\right) + 2 y = 12$

$\frac{1}{2} \cdot 6 + \frac{1}{2} \cdot 2 y + 2 y = 12$
simplifying we get

$3 + y + 2 y = 12$

$3 + 3 y = 12$
so

$3 y = 9$

$y = 3$

so $x = 6 + 2 \cdot 3 = 6 + 6 = 12$

$x = 12 , \setminus y = 3$

#### Explanation:

The given linear equations

$\frac{1}{2} x + 2 y = 12 \setminus \ldots \ldots \ldots . \left(1\right)$

$x - 2 y = 6 \setminus \ldots \ldots \ldots . \left(2\right)$

substituting the value of $x = 6 + 2 y$ from (2) in (1) as follows

$\frac{1}{2} \left(6 + 2 y\right) + 2 y = 12$

$3 + y + 2 y = 12$

$3 y = 9$

$y = 3$

substituting $y = 3$ in (1) we get

$\frac{1}{2} x + 2 \left(3\right) = 12$

$\frac{1}{2} x + 6 = 12$

$\frac{1}{2} x = 6$

$x = 12$

Jul 10, 2018

The solution is the point $\left(12 , 3\right)$.

#### Explanation:

Solve system of equations:

Equation 1: $\frac{1}{2} x + 2 y = 12$

Equation 2: $x - 2 y = 6$

Solve Equation 1 for $x$.

$\frac{1}{2} x + 2 y = 12$

Multiply both sides by $2$.

$x + 4 y = 24$

Subtract $4 y$ from both sides.

$x = 24 - 4 y$

Substitute $24 - 4 y$ for $x$ in Equation 2 and solve for $y$.

$x - 2 y = 6$

$24 - 4 y - 2 y = 6$

Simplify.

$24 - 6 y = 6$

Subtract $24$ from both sides.

$- 6 y = 6 - 24$

Simplify.

$- 6 y = - 18$

Divide both sides by $- 6$.

$y = \frac{- 18}{- 6}$

$y = 3$

Substitute $3$ for $y$ in Equation 1.

$\frac{1}{2} x + 2 y = 12$

$\frac{1}{2} x + 2 \left(3\right) = 12$

$\frac{1}{2} x + 6 = 12$

Subtract $6$ from both sides.

$\frac{1}{2} x = 12 - 6$

Simplify.

$\frac{1}{2} x = 6$

Multiply both sides by $2$.

$x = 12$

The solution is the point $\left(12 , 3\right)$.

graph{(1/2x+2y-12)(x-2y-6)=0 [3.5, 23.5, -3.2, 6.8]}