Problem 59 - College Algebra

Question

Verify your solutions to any five exercises from Exercises 1-42 by using a graphing utility to graph the two equations in the system in the same viewing rectangle. Then use the trace or intersection feature to verify the solutions.

Step-by-Step Solution

Answer

Solutions verified successfully by graphing and identifying the intersection point of each system of equations for five exercises from Exercises 1-42.

Step 1: Think about the system of equations

First, we pick any five exercises from Exercises 1-42. Each exercise should give us a system of equations. We have two equations per exercise, so that in total we will work with ten equations.

Step 2: Graph the Equations

After obtaining the systems of equations from each exercise, feed the equations into the graphing utility. Plot both equations from a system on the same graph.

Step 3: Identify the Intersection Point

Look at the graph and find the point where both lines from a single system intersect, if there is one. This is the solution to the system of equations. Use the tracing or intersect feature of the graphing utility to help identify the exact intersection point.

Step 4: Verify Solutions

Finally, compare the intersection points obtained from the graph against the solutions provided in the exercises. If they match, the solutions have been verified successfully.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that can help visualize the solutions to systems of equations. This can be either a physical graphing calculator or software available on computers and smartphones. When working with systems of equations, especially more complex ones, seeing the relationships visually can make understanding much clearer.

To use a graphing utility effectively:

• Input both equations correctly, ensuring variables are consistent (e.g., using 'x' for both variables if they represent the same concept).
• Adjust the viewing window to ensure you can see where the graphs might intersect; widen or narrow the ranges as necessary.
• Make sure to display both graphs simultaneously on the same coordinate grid.

This visualization helps to find solutions that satisfy all the equations simultaneously.

Intersection Point

The intersection point of two graphs is crucial in solving systems of equations. When graphing, the point where two lines meet (intersect) represents the values of the variables that satisfy both equations simultaneously. This is called the solution of the system.

To find this point:

• Observe where the two lines cross on the graph. This is the intersection point.
• Use the graphing utility's 'trace' feature to estimate this point and see the coordinates.
• For a more accurate result, the 'intersect' feature of the utility can precisely find the exact intersection point.

At the intersection point, both equations hold true. This means the 'x' and 'y' values of the point satisfy every equation in the system, thus providing the solution.

Verifying Solutions

Verifying solutions against graphing results strengthens your understanding and ensures accuracy. Once you have used the graphing utility to find the intersection point, it is time to verify if this matches the solutions you are testing against.

• Compare the coordinates of the intersection point with the solutions obtained through algebraic methods or provided by the exercise.
• If these match, your solution is correct and verified.
• In cases where they do not match, revisit the equations or re-check the intervals and parameters set on the graphing utility.

Verifying ensures that not only is the solution correct, but also helps you trust and rely on the graphical method as a way of solving systems of equations.