Question

Use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places. $$ \left\\{\begin{array}{r} x^{2}+y^{3}=2 \\ x^{3} y=4 \end{array}\right. $$

Short answer

The solution(s) can be determined by identifying the intersection points of the two graphs rounded to two decimal places.

Step-by-step solution

Define the Equations

Identify the two equations provided in the system: 1. \(x^{2} + y^{3} = 2\) 2. \(x^{3} y = 4\)

Choose a Graphing Utility

Select a graphing utility to use, such as a graphing calculator or an online graphing tool like Desmos, WolframAlpha, or GeoGebra.

Input the First Equation

Enter the first equation \(x^{2} + y^{3} = 2\) into the graphing utility. Make sure to properly format the equation so it is recognized correctly by the tool.

Input the Second Equation

Enter the second equation \(x^{3} y = 4\) into the graphing utility. Ensure it is correctly recognized by the tool.

Identify the Intersection Points

Determine where the two graphs intersect. These points of intersection represent the solutions to the system of equations.

Read and Round the Coordinates

Obtain the coordinates of the intersection points from the graphing utility and round them to two decimal places.

Key concepts

graphing utility

When solving systems of equations, a graphing utility is a powerful tool. Graphing utilities can be physical devices like graphing calculators or online platforms such as Desmos, WolframAlpha, or GeoGebra. These tools help visualize the equations by plotting their graphs on a coordinate plane. This visualization helps identify where the graphs intersect, which corresponds to the solution of the system.

To use a graphing utility efficiently:

• Input each equation carefully to ensure the graphing utility recognizes them correctly.
• Use the correct syntax and format as required by the tool.
• Familiarize yourself with the graphing utility interface for smoother navigation and interpretation of results.

Taking these steps will make it easier to find the solutions to your system of equations.

intersection points

Intersection points on a graph are crucial in solving systems of equations. When two equations are graphed, the points where they overlap or intersect represent the solution to the system. This is because these points satisfy both equations simultaneously.

For example, consider the given system of equations:
1. \(x^{2} + y^{3} = 2\)
2. \(x^{3} y = 4\)

Plotting both equations using a graphing utility will show their curves. The points where these curves cross each other are the intersection points. These points are where the values of \(x\) and \(y\) satisfy both equations.

Identifying intersection points:

• Observe where the graphs meet on the coordinate plane.
• Use the graphing utility's features to highlight or mark these points.
• Note the coordinates, which represent the solutions to the system.

Mastering this concept helps significantly in graphically solving complex systems of equations.

rounding coordinates

Once you've identified the intersection points using a graphing utility, an important next step is rounding the coordinates. Rounding ensures that your solutions are practical and presentable, especially when dealing with decimal values.

In the given problem, you are instructed to round the coordinates of the intersection points to two decimal places:

• After identifying the intersection points, take note of the exact coordinates.
• Round each value to the nearest hundredth (two decimal places). For example, if an intersection point is (2.456, 3.789), you would round it to (2.46, 3.79).
• Ensure consistency in rounding to maintain the precision of your solutions.

Rounding coordinates helps simplify your results while keeping them accurate. It's a key step in making mathematical solutions clear and digestible, especially in educational contexts.