Q 9 - Intermediate Algebra

Question

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

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

Step-by-Step Solution

Answer

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

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

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

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