Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{array}{l} x-y^{2}=-2 \\ x-2 y=6 \end{array}\right.$$

Step-by-Step Solution

Answer

Check the points of intersection on the graph and round the values to three decimal points. Then verify by substituting these values into the original equations. The exact solutions will depend on the graph plotted and how finely you read off the points.

Step 1: Graph the Equations

First, use a graphing tool (like Desmos, Grapher or another preferred calculator) to draw the two graphs expressed by these equations. Using a graphing utility is efficient for visualizing and approximating solutions for systems of equations. The first equation can be solved for x, resulting in \\(\begin{equation}x = y^{2} + 2 \\end{equation}\\). The second equation can also be solved for x, resulting in \\(\begin{equation}x = 2y + 6 \\end{equation}\\). When you graph these two equations, you will see the points at which the lines intersect, these points are the approximate solutions to the system of equations.

Step 2: Find the Intersection Points

Identify the intersection points of the two curves on your graph. These intersections represent the visual solutions to the system of equations. Record these points, by rounding the x and y coordinates to three decimal places.

Step 3: Verify the Solutions

Now, the found solutions need to be checked by plugging them back into each of the original equations. If plugging the x- and y-values of the solutions into the equations makes both equations true, then the solutions are correct.

Key Concepts

Graphing Utilities

Graphing utilities are incredibly helpful tools when dealing with systems of equations. They allow you to visualize the equations and their intersections on a graph. In this scenario, the two given equations can be graphed to find where they intersect, revealing potential solutions. By graphing the equations first, we can see where they might meet without having to perform extensive manual calculations.

• Accuracy: Graphing utilities compute points accurately and clearly, reducing human error.
• Ease of Use: Tools like Desmos or Grapher are user-friendly and offer intuitive interfaces.
• Visualization: Seeing the equations graphically can help deepen understanding of the solution process.

For the provided system of equations, we first rearranged each to solve for \(x\):- First equation: \(x = y^2 + 2\)- Second equation: \(x = 2y + 6\)Using a graphing utility, you input these equations to generate a visual representation. The intersection points will give you an initial estimation of the solutions to the system.

Intersection Points

Finding intersection points is crucial in solving systems of equations graphically. These points show where the graphs of the equations cross each other, providing the solutions to the system. By determining the coordinates of these intersections, you can check for the solution set of the equations.Once you've graphed the two equations, look for the crossing points. These are your intersection points. For this specific system:- The graph of \(x = y^2 + 2\) is a parabola.- The graph of \(x = 2y + 6\) is a straight line.Where these two shapes meet on the graph are the locations of your intersection points, which should be identified and rounded to three decimal places for accuracy.

• Visual method: Easily spot the intersection directly on the graph.
• Numeric method: Use the utility's coordinate display to get precise coordinates.
• Approximation: Since graphing utilities deal with visual data, they provide approximate values for intersection points, which require rounding.

By rounding to three decimal places, you ensure precision while maintaining simplicity in your calculations.

Verification of Solutions

Verification is a crucial step. It ensures that the solutions you obtained are correct by substituting them back into the original equations. Verification checks whether the solutions make both equations true simultaneously.Once intersection points are found from the graph, use these points to verify by plugging them back into the original equations:1. Substitute the \(x\) and \(y\) value of the first intersection point into both equations: - Does it satisfy \(x - y^2 = -2\)? - Does it satisfy \(x - 2y = 6\)?2. Repeat for any additional intersection points.

• Consistency Check: Ensures the points work for the original conditions.
• Prevention of Error: Eliminates false solutions derived from potential graphing discrepancies.

If the intersection points satisfy both equations, they are confirmed as solutions to the system. This step is crucial to ensure the correctness of the results obtained through graphical methods.