Question

In the following exercises, solve the following system of equations by graphing

$$\begin{gathered} 3x+y=6 \\ x+3y=-6 \end{gathered}$$

Short answer

The solution to the linear equations is (3,-3).

Step-by-step solution

Step 1. Given information

The given linear equations are

$$\begin{gathered} 3x+y=6 \\ x+3y=-6 \end{gathered}$$

Step 2. Finding the intersecting 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.

$$\begin{gathered} \mathrm{Solve}\mathrm{the}\mathrm{first}\mathrm{equation}\mathrm{for}\mathrm{y}. \\ \mathrm{and}\mathrm{find}\mathrm{the}\mathrm{slope}\mathrm{and}\mathrm{y}-\mathrm{intercept}. \\ 3\mathrm{x}+\mathrm{y}=6 \\ \mathrm{y}=-3\mathrm{x}+6 \\ \mathrm{Here}\mathrm{the}\mathrm{slope}\mathrm{is}\mathrm{m}=-3 \\ \mathrm{And}\mathrm{the}\mathrm{y}-\mathrm{intercept}\mathrm{is}\mathrm{b}=6 \end{gathered}$$

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

$$\begin{gathered} x+3y=-6 \\ 3y=-x-6 \\ y=-\frac{1}{3}x-2 \\ \mathrm{Here}\mathrm{the}\mathrm{slope}\mathrm{is}m=-\frac{1}{3} \\ \mathrm{And}\mathrm{the}\mathrm{y}-\mathrm{intercept}\mathrm{is}b=-2 \end{gathered}$$$$\begin{gathered} x+3y=-6 \\ 3y=-x-6 \\ y=-\frac{1}{3}x-2 \\ \mathrm{Here}\mathrm{the}\mathrm{slope}\mathrm{i}sm=-\frac{1}{3} \\ \mathrm{And}\mathrm{the}\mathrm{y}-\mathrm{intercept}\mathrm{is}b=-2 \end{gathered}$$

Step 3. The graph obtained

$$\begin{gathered} \mathrm{Check}: \\ \mathrm{First}\mathrm{substitute}\mathrm{x}=3, \\ \mathrm{y}=-3\mathrm{into}\mathrm{the}\mathrm{equation} \\ 3x+y=6 \\ 3x+y=6 \\ 3\left(3\right)-3=6 \\ 9-3=6 \\ 6=6 \\ \mathrm{This}\mathrm{is}\mathrm{true}. \\ \mathrm{Also}\mathrm{substitute}\mathrm{x}=3,\mathrm{y}=-3 \\ \mathrm{into}\mathrm{the}\mathrm{equatio}nx+3y=-6 \\ x+3y=-6 \\ 3+3(-3)=-6 \\ 3-9=-6 \\ -6=-6\mathrm{This}\mathrm{is}\mathrm{true}. \end{gathered}$$