Problem 60 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to find the point(s) of intersection of the graphs. Then confirm your solution algebraically. $$\left\\{\begin{array}{l}x-y=3 \\ x-y^{2}=1\end{array}\right.$$

Step-by-Step Solution

Answer

The intersection points obtained graphically, should be validated algebraically. Only the points which satisfy both equations are the true solutions.

Step 1: Graph the equations

Use a graphing utility to sketch the graphs of the equations \(x - y = 3\) and \(x - y^2 = 1\). The points where the graphs intersect are the solutions of the equations.

Step 2: Identifying the Intersection Points

Identify the intersection points from the graph. These will be the potential solutions.

Step 3: Algebraic Confirmation

Plug the coordinates of the intersection points into both equations to confirm if they are indeed the solutions. A valid solution should satisfy both equations.

Key Concepts

Graphing Utility

A graphing utility is an electronic tool or software that helps visualize mathematical equations with ease. It's an incredibly useful tool for solving systems of equations by providing a visual representation of the situation.
Graphing utilities can plot lines, curves, and other equations. This visual aspect allows you to see where two or more graphs intersect, aiding in finding solutions to the equations.

• By inputting the equations into the graphing utility, you can instantly see the plotted graphs on the coordinate plane.
• Each line or curve on the graph represents one of the equations in your system.
• The points of intersection indicate potential solutions to the system as these points provide common solutions to all the equations involved.

Such utilities commonly come with features like zooming, tracing, and calculating intersections directly. This makes it a go-to method for students to quickly and accurately find intersections before diving into algebraic solutions.

Intersection Points

Intersection points are crucial in solving systems of equations graphically. Let's break down why they matter and how to find them:
When two graphs intersect, it means that at those points, both equations share the same values for x and y. In simpler terms, they are the solutions to both equations. Finding these points is usually the goal when solving systems graphically.

• Use the graphing utility to plot the provided equations.
• Look for points where the lines or curves cross each other.
• Each crossing point is a potential solution to the system.

Identifying intersection points visually is generally straightforward, but accuracy is important. Once identified, you can double-check these points with algebraic methods to confirm that they satisfy all equations in the system.

Algebraic Solution

After finding possible intersection points graphically, the next step is to confirm these solutions algebraically. This involves plugging the coordinates of the candidates into both given equations and checking if they hold true.

• Take each intersection point found from the graph.
• Substitute the x and y values into each of the original equations.
• If both equations are satisfied (i.e., they balance), the point is a confirmed solution to the system.

For the given example, first, substitute the intersection point into the equation \( x - y = 3 \). Ensure it yields a true statement. Then, check it against the equation \( x - y^2 = 1 \). When both checks are successful, you have confirmed your solution algebraically. This step solidifies the graphical findings and ensures mathematical rigor and accuracy in your solution.