Question

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

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

Short answer

The solution of the system of linear equations is (2,4).

Step-by-step solution

Step 1. Given information  

Two linear equations are

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

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.

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

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

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

Step 3. Graph obtained  

Step 4. Checking the equations

$$\begin{gathered} \mathrm{First}\mathrm{substitute}x=2, \\ y=4 \\ \mathrm{into}\mathrm{the}\mathrm{equation}y=\frac{3}{2}x+1 \\ y=\frac{3}{2}x+1 \\ 4=\frac{3}{2}\left(2\right)+1 \\ 4=3+1 \\ 4=4 \\ \mathrm{This}\mathrm{is}\mathrm{true}. \end{gathered}$$

$$\begin{gathered} \mathrm{First}\mathrm{substitute}x=2, \\ y=4 \\ \mathrm{into}\mathrm{the}\mathrm{equation}y=\frac{3}{2}x+1 \\ y=\frac{3}{2}x+1 \\ 4=\frac{3}{2}\left(2\right)+1 \\ 4=3+1 \\ 4=4 \\ \mathrm{This}\mathrm{is}\mathrm{true}. \end{gathered}$$