Question

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

$$\begin{gathered} x+4y=12 \\ -x+y=3 \end{gathered}$$

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

$$\begin{gathered} x+4y=12 \\ -x+y=3 \end{gathered}$$

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

The first equation is

$x+4y=12$

Solve for $y$

$$\begin{gathered} x+4y=12 \\ x+4y-x=12-x \\ 4y=-x+12 \\ \frac{4y}{4}=-\frac{1}{4}x+\frac{12}{4} \\ y=-\frac{1}{4}x+3 \end{gathered}$$

Compare with the slope-intercept formula of line

$y=mx+b$

So,

$$\begin{gathered} m=-\frac{1}{4} \\ b=3 \end{gathered}$$

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

The second equation is

$-x+y=3$

Solve for $y$

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

Compare with the slope-intercept formula of line

$y=mx+b$

So,

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

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

First equation:

Slope is $-\frac{1}{4}$

y-intercept is $3$

Second equation:

Slope is $1$

y-intercept is $3$

Since the slopes are different, the lines intersect.

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