Chapter 9.9, Problem 2ST

(a)

To determine

To graph: the given equations.

Answer to Problem 2ST

  

Explanation of Solution

Given:

  $\begin{array}{l}{x}^2=4y\\ xy=8\end{array}$

Calculation:

The given system of equations is:

  $\begin{array}{l}{x}^2=4y\\ xy=8\end{array}$

The graph can be shown as below:

  

Conclusion:

  

(b)

To determine

the number of real solutions.

Answer to Problem 2ST

Number of real solutions is one.

Explanation of Solution

Calculation:

From the above graph:

The two graphs intersected in only one point.

Therefore, the number of real solutions of the system is one

Conclusion:

Number of real solutions is one.

(c)

To determine

To estimate: each real solution to the nearest half-unit.

Answer to Problem 2ST

Real solution is $\left(3.175,2.52\right)$ .

Explanation of Solution

Calculation:

From the given system of equations as below:

  $\begin{array}{l}{x}^2=4y\\ xy=8\end{array}$

Rewriting the second equation,

  $\begin{array}{l}xy=8\\ y=\frac{8}{x}\end{array}$

Substituting $y=\frac{8}{x}$ in the first equation and solving the resulting equation,

  $\begin{array}{l}{x}^2=4y\\ x^2=4\left(\frac{8}{x}\right)\left[Substitutingy=\frac{8}{x}\right]\\ x^3=32\\ x={32}^{\frac{1}{3}}\\ =3.175\end{array}$

Now substituting x value in $y=\frac{8}{x}$ ,

  $\begin{array}{l}\text{If }x=3.175\\ y=\frac{8}{x}\\ =\frac{8}{\left(3.175\right)}\left[\text{Replacing }x=3.175\right]\\ =2.52\end{array}$

Therefore, real solution is $\boxed{\left(3.175,2.52\right)}$ .

Conclusion:

Real solution is $\left(3.175,2.52\right)$ .