Question

Solve each system by elimination: $\left\{\begin{array}{l}\frac{1}{3}x-\frac{1}{2}y=1\\ \frac{3}{4}x-y=\frac{5}{2}\end{array}\right.$

Short answer

The solution of the system are$$\begin{gathered} x=6 \\ y=2 \end{gathered}$$.

Step-by-step solution

Given information.

The given system of equations is,

$\left\{\begin{array}{l}\frac{1}{3}x-\frac{1}{2}y=1\\ \frac{3}{4}x-y=\frac{5}{2}\end{array}\right.$

Isolation of variable

The equation is $\frac{1}{3}x-\frac{1}{2}y=1$. Isolating the variable. We get $x=3+\frac{3}{2}y$.

Now substituting the value of $x$in the other equation, we get

$\left[\frac{3}{4}\left(3+\frac{3}{2}y\right)-y=\frac{5}{2}\right]$.

Isolating and solving further

Further solving, we get$\left[\frac{18+y}{8}=\frac{5}{2}\right]$, isolating $y$, we get

$y=2$

Substitute,$y=2$, we get

$x=3+\frac{3}{2}·2$

$x=6$

So the values are$$\begin{gathered} x=6 \\ y=2 \end{gathered}$$

Checking the value

Substituting, we get

$$\begin{gathered} \frac{1}{3}\left(6\right)-\frac{1}{2}\left(2\right)=1 \\ 2-1=1 \\ 1=1 \end{gathered}$$

Checking for another value

Checking for another value, we get

$$\begin{gathered} \frac{3}{4}\left(6\right)-2=\frac{5}{2} \\ \frac{9}{2}-2=\frac{5}{2} \\ \frac{5}{2}=\frac{5}{2} \end{gathered}$$

So the value is correct