Question

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

$$\begin{gathered} y=\frac{2}{3}x-2 \\ y=-\frac{1}{3}x-5 \end{gathered}$$

Short answer

The solution of the system of linear equations is (-3,-4).

Step-by-step solution

Step 1. Given information  

Two linear equations are

$$\begin{gathered} y=\frac{2}{3}x-2 \\ y=-\frac{1}{3}x-5 \end{gathered}$$

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.

$$\begin{gathered} y=\frac{2}{3}\times x-2 \\ \mathrm{Here}\mathrm{the}\mathrm{slope}\mathrm{is}m=\frac{2}{3} \\ \mathrm{And}\mathrm{the}\mathrm{y}-\mathrm{intercept}\mathrm{is}b=-2 \end{gathered}$$

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

$$\begin{gathered} y=-\frac{1}{3}\times x-5 \\ \mathrm{Here}\mathrm{the}\mathrm{slope}\mathrm{is}m=-1/3 \\ \mathrm{And}\mathrm{the}\mathrm{y}-\mathrm{intercept}\mathrm{is}b=-5 \end{gathered}$$

Step 3. Graph obtained  

Step 4. Checking the equations 

$$\begin{gathered} \mathbf{Check}\mathbf{:} \\ \mathrm{First}\mathrm{substitute}x=-3, \\ y=-4\mathrm{into}\mathrm{the}\mathrm{equation} \\ y=\frac{2}{3}x-2 \\ -4=\frac{2}{3}(-3)-2 \\ -4=-2-2 \\ -4=4 \\ \mathrm{This}\mathrm{is}\mathrm{true}. \end{gathered}$$

$$\begin{gathered} \mathrm{Also}\mathrm{substitute}x=-3, \\ y=-4\mathrm{into}\mathrm{the}\mathrm{equation} \\ y=-\frac{1}{3}\times x-5 \\ y=-\frac{1}{3}\times x-5 \\ -4=-\frac{1}{3}\times (-3)-5 \\ -4=1-5 \\ -4=4 \\ \mathrm{This}\mathrm{is}\mathrm{true}. \end{gathered}$$