Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions,

express them in the ordered-pair form given in Example 6.

$\begin{array}{l}\frac{3}{2}x-\frac{1}{3}y=\frac{1}{2}\\ 2x-\frac{1}{2}y=-\frac{1}{2}\end{array}$

Short answer

Hence, the solution of the system of equations $\begin{array}{l}\frac{3}{2}x-\frac{1}{3}y=\frac{1}{2}\\ 2x-\frac{1}{2}y=-\frac{1}{2}\end{array}$is the ordered pair

Step-by-step solution

Step 1. Given information.

Given a system of equations$\begin{array}{l}\frac{3}{2}x-\frac{1}{3}y=\frac{1}{2}\\ 2x-\frac{1}{2}y=-\frac{1}{2}\end{array}$

substitution method.

Step 2. Write down the concept.

Given equation,

$\frac{3}{2}x-\frac{1}{3}y=\frac{1}{2}$ …….(1)

$2x-\frac{1}{2}y=-\frac{1}{2}$ …….(2)

Solving equation (1) for x,

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

Step 3. Determining the angle

We are substituting equation (3) in equation (2),

$2\left(\frac{1}{3}+\frac{2}{9}y\right)-\frac{1}{2}y=-\frac{1}{2}$

$\frac{2}{3}+\frac{4}{9}y-\frac{1}{2}y=-\frac{1}{2}$

Subtracting $\frac{2}{3}$ from both sides,

$\begin{array}{l}\frac{4}{9}y-\frac{1}{2}y=-\frac{1}{2}-\frac{2}{3}\\ \frac{8y-9y}{18}=\frac{-3-4}{6}\\ \frac{-y}{18}=\frac{-7}{6}\end{array}$

Multiplying both sides by -18

$y=21$

Now, substituting back value of y in equation (3)

$\begin{array}{l}x=\frac{1}{3}+\frac{2}{9}\left(21\right)\\ x=\frac{1}{3}+\frac{14}{3}\\ x=5\end{array}$