How do you solve the system 1/2x+2y=12 and x-2y=6 using substitution?

May 3, 2017

See the entire solution process below:

Explanation:

Step 1) Solve the second equation for $x$:

$x - 2 y = 6$

$x - 2 y + \textcolor{red}{2 y} = 6 + \textcolor{red}{2 y}$

$x - 0 = 6 + 2 y$

$x = 6 + 2 y$

Step 2) Substitute $6 + 2 y$ for $x$ in the first equation and solve for $y$:

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

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

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

$\frac{6}{2} + \left(\frac{1}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}}} \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} y\right) + 2 y = 12$

$3 + 1 y + 2 y = 12$

$3 + \left(1 + 2\right) y = 12$

$3 + 3 y = 12$

$- \textcolor{red}{3} + 3 + 3 y = - \textcolor{red}{3} + 12$

$0 + 3 y = 9$

$3 y = 9$

$\frac{3 y}{\textcolor{red}{3}} = \frac{9}{\textcolor{red}{3}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} y}{\cancel{\textcolor{red}{3}}} = 3$

$y = 3$

Step 3) Substitute $3$ for $y$ in the solution to the second equation at the end of Step 1 and calculate $x$:

$x = 6 + 2 y$ becomes:

$x = 6 + \left(2 \cdot 3\right)$

$x = 6 + 6$

$x = 12$

The solution is: $x = 12$ and $y = 3$ or $\left(12 , 3\right)$