Chapter 7.1, Problem 51E

To determine

Tofind:The intersection point of the system of equations 3 $\{\begin{array}{c}{x}^2+y^2=8\\ y=x^2\end{array}$ using graphical utility and verify the result by checking them in the original system.

Answer to Problem 51E

The intersection points of the system of equations $\{\begin{array}{c}{x}^2+y^2=8\\ y=x^2\end{array}$ are $\left(1.540,2.372\right)$ and $\left(-1.540,2.372\right)$ .

Explanation of Solution

Given information:

The given system of equations is $\{\begin{array}{c}{x}^2+y^2=8\\ y=x^2\end{array}$ .

Calculation:

Draw the graph for the equations $x^2+y^2=8$ and $y=x^2$ .

  

From the above graph, it can be seen that the intersection point is $\left(1.540,2.372\right)$ and $\left(-1.540,2.372\right)$ . So, the solution of the system of equations $\{\begin{array}{c}{x}^2+y^2=8\\ y=x^2\end{array}$ is $\left(1.540,2.372\right)$ and $\left(-1.540,2.372\right)$ .

Verify the solution by substituting the values of intersection point in the system of equations.

Substitute $1.540$ for $x$ and $2.372$ for $y$ in the first equation.

  $\begin{array}{c}{x}^2+y^2=8\\ {\left(1.540\right)}^2+{\left(2.372\right)}^2=8\\ 2.3716+5.6263=8\\ 8=8\end{array}$

Substitute $1.540$ for $x$ and $2.372$ for $y$ in the second equation.

  $\begin{array}{c}y=x^2\\ 2.372={\left(1.540\right)}^2\\ 2.372=2.372\end{array}$

Now, it is clear that the intersection point satisfied both the equations.So, the ordered pair $\left(1.540,2.372\right)$ is a solution of the system of equations.

Substitute $-1.540$ for $x$ and $2.372$ for $y$ in the first equation.

  $\begin{array}{c}{x}^2+y^2=8\\ {\left(-1.540\right)}^2+{\left(2.372\right)}^2=8\\ 2.3716+5.6263=8\\ 8=8\end{array}$

Substitute $-1.540$ for $x$ and $2.372$ for $y$ in the second equation.

  $\begin{array}{c}y=x^2\\ 2.372={\left(-1.540\right)}^2\\ 2.372=2.372\end{array}$

Now, it is clear that the intersection point satisfied both the equations.So, the ordered pair $\left(-1.540,2.372\right)$ is a solution of the system of equations.

Therefore, the intersection points of the system of equations $\{\begin{array}{c}{x}^2+y^2=8\\ y=x^2\end{array}$ are $\left(1.540,2.372\right)$ and $\left(-1.540,2.372\right)$ .