Question

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

$$\begin{gathered} 2x-y=4 \\ 2x+3y=12 \end{gathered}$$

Short answer

The solution of the system of linear equations is (3,2).

Step-by-step solution

Step 1. Given information

The linear equations are

$$\begin{gathered} 2x-y=4 \\ 2x+3y=12 \end{gathered}$$

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.

$$\begin{gathered} 2x-y=4 \\ -y=-2x+4 \\ y=2x-4 \\ \mathrm{Here}\mathrm{the}\mathrm{slope}\mathrm{is}m=2 \\ \mathrm{And}\mathrm{the}\mathrm{y}-\mathrm{intercept}\mathrm{is}b=-4 \end{gathered}$$

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

$$\begin{gathered} 2x+3y=12 \\ 3y=-2x+12 \\ y=-\frac{2}{3}\times x+\frac{12}{3} \\ y=-\frac{2}{3}x+4 \\ \mathrm{Here}\mathrm{the}\mathrm{slope}\mathrm{is}m=-\frac{2}{3} \\ \mathrm{And}\mathrm{the}\mathrm{y}-\mathrm{intercept}\mathrm{is}b=4 \end{gathered}$$

Step 3. Graph obtained

The solution of the system of linear equations is (3,2).

Step 4. Checking the equations

$$\begin{gathered} \mathrm{First}\mathrm{substitute}x=3, \\ y=2\mathrm{into}\mathrm{the}\mathrm{equation} \\ 2x-y=4 \\ 2\left(3\right)-2=4 \\ 6-2=4 \\ 4=4 \\ \mathrm{This}\mathrm{is}\mathrm{true}. \end{gathered}$$

$$\begin{gathered} \mathrm{Also}\mathrm{substitute}x=3, \\ y=2\mathrm{into}\mathrm{the}\mathrm{equation} \\ 2x+3y=12 \\ 2x+3y=12 \\ 2\left(3\right)+3\left(2\right)=12 \\ 6+6=12 \\ 12=12 \\ \mathrm{This}\mathrm{is}\mathrm{true}. \end{gathered}$$