Chapter 10.8, Problem 39E

To determine

To find: solutions of the system of equations using the graphical method

Answer to Problem 39E

The solutions of the system of equations are $\left(-2.30,-0.70\right)$ and $\left(0.48,-1.19\right)$

Explanation of Solution

Given:

  $\{\begin{array}{c}{x}^4+16y^4=32\\ x^2+2x+y=0\end{array}$

Calculation:

Consider the system of equations.

  $\{\begin{array}{c}{x}^4+16y^4=32\\ x^2+2x+y=0\end{array}$

In order to solve the system in graphing method, solve for y in terms of x.

  $x^4+16y^4=32$

  $16y^4=-x^4+32$ Subtract $x^4$

  $y^4=-\frac{1}{16}{x}^4+2$ Divide by 16 on both sides

  $y={\left(-\frac{1}{16}{x}^4+2\right)}^{\frac{1}{4}}$ Take $4^{\text{th}}$ root on both sides

And the second equation is,

  $x^2+2x+y=0$

  $y=-\left(x^2+2x\right)$ subtract $\left(x^2+2x\right)$ on both sides

The system of equations becomes

  $\{\begin{array}{l}y={\left(-\frac{1}{16}{x}^4+2\right)}^{\frac{1}{4}}\\ y=-\left(x^2+2x\right)\end{array}$

Find the intersect points:

Enter the equations in graphing calculator as follows.

  

Set the windows as follows.

  

To get the graph of the equations, press GRAPH button.

  

To find the intersection points, press $2^{nd}+\text{Trace}$ then Press 5.

Then confirm the first and second curves by using the right and left buttons.

  

Find the other intersect point.

  

From the screen shots, the intersection points are,

  $\left(-2.30,-0.70\right)$ and $\left(0.48,-1.19\right)$

  

Hence, the solutions of the system of equations are $\left(-2.30,-0.70\right)$ and $\left(0.48,-1.19\right)$

Conclusion:

Hence, the solutions of the system of equations are $\left(-2.30,-0.70\right)$ and $\left(0.48,-1.19\right)$