Question

In the following exercises, solve the following systems of equations by graphing.

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

Short answer

The solution of the system of linear equations is (0,1).

Step-by-step solution

Step 1. Given information

Two linear equations are

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

Step 2. Solving equations for the intersection points 

First, solve both of these equations for y such that their slopes and y intercepts may be easily graphed.

And find the slope and y-intercept by solving the first equation for y.

$$\begin{gathered} -x+3y=3 \\ 3y=x+3 \\ y=\frac{1}{3}\times x+\frac{3}{3} \\ y=\frac{1}{3}\times x+1 \\ \mathrm{Here}\mathrm{the}\mathrm{slope}\mathrm{is}m=\frac{1}{3} \\ \mathrm{And}\mathrm{the}\mathrm{y}-\mathrm{intercept}\mathrm{is}b=1 \end{gathered}$$

Again find the slope and y-intercept by solving the Second equation for y.

$$\begin{gathered} x+3y=3 \\ 3y=-x+3 \\ y=-\frac{1}{3}\times x+\frac{3}{3} \\ y=-\frac{1}{3}\times x+1 \\ \mathrm{Here}\mathrm{the}\mathrm{slope}\mathrm{is}m=-\frac{1}{3} \\ \mathrm{And}\mathrm{the}\mathrm{y}-\mathrm{intercept}\mathrm{is}b=1 \end{gathered}$$

localid="1644480436730" $$\begin{gathered} x+3y=3 \\ 3y=-x+3 \\ y=-\frac{1}{3}\times x+3/3 \\ y=-\frac{1}{3}\times x+1 \\ \mathrm{Here}\mathrm{the}\mathrm{slope}\mathrm{is}m=-\frac{1}{3} \\ \mathrm{And}\mathrm{the}\mathrm{y}-\mathrm{intercept}\mathrm{is}b=1 \end{gathered}$$

Step 3. Graph obtained

$$\begin{gathered} \mathrm{Check}: \\ \mathrm{First}\mathrm{substitute}x=0, \\ y=1\mathrm{into}\mathrm{the}\mathrm{equation} \\ -x+3y=3 \\ 0+3=3 \\ 3=3 \\ \mathrm{This}\mathrm{is}\mathrm{true}. \end{gathered}$$

$$\begin{gathered} \mathrm{Also}\mathrm{substitute}x=0, \\ y=1\mathrm{into}\mathrm{the}\mathrm{equation} \\ x+3y=3 \\ x+3y=3 \\ 0+3\left(1\right)=3 \\ 0+3=3 \\ 3=3 \\ \mathrm{This}\mathrm{is}\mathrm{true}. \end{gathered}$$

The solution of the system of linear equations is (0,1)

$$\begin{gathered} \mathrm{Check}: \\ \mathrm{First}\mathrm{substitute}x=0, \\ y=1\mathrm{into}\mathrm{the}\mathrm{equation} \\ x+y=-4 \\ -2-2=-4 \\ -4=4 \\ \mathrm{This}\mathrm{is}\mathrm{true}. \end{gathered}$$