Question

Without graphing, determine the number of solutions and then classify the system of equations.

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

Short answer

The number of solutions is 0 and the system of equations are parallel lines, inconsistent and independent.

Step-by-step solution

Step 1. Given information.

The system of equations is$\left\{\begin{array}{l}y=\frac{2}{3}x+1\\ -2x+3y=5\end{array}\right.$.

Step 2. Solve the first equation for y.

Find the slope andy-intercept.

$y=\frac{2}{3}x+1$

Hence the slope is $\frac{2}{3}$and the y-intercept is 1.

Step 3. Solve the second equation for y.

Find the slope and y-intercept.

$$\begin{gathered} -2x+3y=5 \\ y=\frac{2}{3}x+\frac{5}{3} \end{gathered}$$

Hence the slope is $\frac{2}{3}$and the y-intercept is $\frac{5}{3}$.

Step 4. Explain the number of solutions. 

• Here the slopes are the same and y-intercepts are different.
• The system of equations are parallel lines, inconsistent and independent and the number of solutions is 0.