Question

In Problems 5–24, graph each equation of the system. Then solve the system to find the points of intersection.

$$\begin{gathered} xy=1; \\ y=2x+1 \end{gathered}$$

Short answer

The graph of the system of equations $xy=1;y=2x+1$is:

Step-by-step solution

Step 1. Given

The system of non-linear equation:

$$\begin{gathered} xy=1; \\ y=2x+1 \end{gathered}$$

To graph the equation and to find the point of intersection.

Step 2. Graph the equations

Graph the equations in the same plane.

Step 3. To find the point of intersection.

$$\begin{gathered} xy=1 \\ y=\frac{1}{x} \end{gathered}$$

Substitute the above value in the second equation,

$$\begin{gathered} \frac{1}{x}=2x+1 \\ 1=2x^2+x \\ 2x^2+x-1=0 \\ (x+1)(2x-1)=0 \\ x=-1,\frac{1}{2} \end{gathered}$$

Step 4. Find y

When $x=-1$,

$$\begin{gathered} y=2x+1 \\ y=2(-1)+1 \\ y=-1 \end{gathered}$$

When $x=\frac{1}{2}$,

$$\begin{gathered} y=2\left(\frac{1}{2}\right)+1 \\ =2 \end{gathered}$$

The points of intersection are $(-1,-1),(\frac{1}{2},2)$

Step 5. Plot the point of intersection

The graph of the system of non-linear equations with the point of intersection is: