Question

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

$$\begin{gathered} 3x+2y=2 \\ 2x+y=1 \end{gathered}$$

Short answer

The system of the equation has 1 solution and is consistent and independent.

Step-by-step solution

Step 1. Given information 

The system of equations is

$$\begin{gathered} 3x+2y=2 \\ 2x+y=1 \end{gathered}$$

Step 2. Find the slope and intercept of the first equation.

The first equation is

$3x+2y=2$

Solve for $y$

$$\begin{gathered} 3x+2y=2 \\ 3x+2y-3x=2-3x \\ 2y=-3x+2 \\ \frac{2y}{2}=-\frac{3}{2}x+\frac{2}{2} \\ y=-\frac{3}{2}x+1 \end{gathered}$$

Compare with the slope-intercept formula of line

$y=mx+b$

So,

$$\begin{gathered} m=-\frac{3}{2} \\ b=1 \end{gathered}$$

Step 3. Find the slope and intercept of the second equation.

The second equation is

$2x+y=1$

Solve for $y$

$$\begin{gathered} 2x+y-2x=1-2x \\ y=-2x+1 \end{gathered}$$

Compare with the slope-intercept formula of line

$y=mx+b$

So,

$$\begin{gathered} m=-2 \\ b=1 \end{gathered}$$

Step 4. Compare the slopes and intercepts of the two lines.

First equation:

Slope is $-\frac{3}{2}$

y-intercept is $1$

Second equation:

Slope is $-2$

y-intercept is $1$

Since the slopes are different, the lines intersect.

A system of equations whose graphs are intersecting has 1 solution and is consistent and independent.