Question

Use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places. $$ \left\\{\begin{array}{r} x^{4}+y^{4}=12 \\ x y^{2}=2 \end{array}\right. $$

Short answer

The solutions are approximately (1.29, 1.24) and (-1.29, -1.24).

Step-by-step solution

Graph the First Equation

Use a graphing utility to plot the first equation, which is \(x^{4} + y^{4} = 12\). This defines a curve in the xy-plane.

Graph the Second Equation

Now, plot the second equation, \(x y^{2} = 2\). This defines another curve in the xy-plane.

Find Intersection Points

Check where the two graphs intersect. These intersection points are the solutions to the system of equations. Zoom in as needed and use the graphing utility's feature to find the exact intersection points.

Round the Solutions

Take the intersection points and round the coordinates to two decimal places.

Key concepts

graphing calculator

A graphing calculator is a powerful tool used to visualize equations and find solutions graphically. For this exercise, we'll use a graphing calculator to plot and solve the system of equations.
Start by inputting the first equation, \(x^{4} + y^{4} = 12\), into the graphing utility. This will show a specific curve in the xy-plane.
Next, input the second equation, \(xy^{2} = 2\), which will display another curve. The graphing calculator can plot both equations simultaneously, making it easier to see where they intersect.
By adjusting the viewing window and zooming in, you can get a clearer view of the intersection points. Use the calculator's feature to pinpoint the exact coordinates.

intersection points

The points where the curves of the two equations intersect are called the intersection points. These points represent the simultaneous solutions to the system of equations.
To locate these points using the graphing calculator, first make sure both equations are plotted on the same graph. Look for areas where the curves meet.
Once you spot these intersections, use the built-in function of the graphing utility to calculate the exact coordinates. This process often involves highlighting the intersection and selecting it to get the precise x and y values. Remember, accurate identification of these points is critical for solving the equations correctly.

rounding

Rounding is essential to present solutions in a simplified manner. After identifying the intersection points, round each coordinate to two decimal places as required for this exercise.
For example, if the intersection point is at (1.2345, -0.9876), you would round it to (1.23, -0.99).
Rounding makes the values easier to work with and is often necessary when providing solutions in real-world scenarios.
Ensure to round consistently and follow specific rounding rules, such as rounding up if the next digit is 5 or more and rounding down if it is less than 5.