Q 330 - Intermediate Algebra

Question

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

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

Step-by-Step Solution

Answer

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

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.  

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

Step 3. The graph obtained

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