Question

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

$\left\{\begin{array}{l}2x-y=4\\ 2x+3y=12\end{array}\right.$

Short answer

The solutions to the system of equations are$x=3$and$y=2.$

Step-by-step solution

Step 1. Given information.

We have:

$\left\{\begin{array}{l}2x-y=4\\ 2x+3y=12\end{array}\right.$

Step 2. Find the intercepts.   

We will find the x-intercept and y-intercept of both equations and use them to graph the lines.

The system of equations are:

$\left\{\begin{array}{l}2x-y=4\\ 2x+3y=12\end{array}\right.$

x-intercept: substitute $y=0$

$$\begin{gathered} 2x-0=4 \\ x=2 \end{gathered}$$

X-intercept: $\left(2,0\right)$

$$\begin{gathered} 2x+3\left(0\right)=12 \\ x=6 \end{gathered}$$

X-intercept: $\left(6,0\right)$

y-intercept: substitute $x=0$

$$\begin{gathered} 2\left(0\right)-y=4 \\ y=-4 \end{gathered}$$

Y-intercept: $\left(0,-4\right)$

$$\begin{gathered} 2\left(0\right)+3y=12 \\ y=4 \end{gathered}$$

Y-intercept:$\left(0,4\right)$

Step 3. Use the intercept to draw the graph.  

Graph the line and find the intersection point:

The lines intersect at $\left(3,2\right)$.

Therefore, the solutions are:

$x=3,y=2$