Problem 52 - 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. $$\left\\{\begin{aligned} x^{2}+y^{2} &=25 \\ (x-8)^{2}+y^{2} &=41 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The two circles intersect at (4.146, 2.646) and (4.146, -2.646).

Step 1: Visual Approximation

First, plot the two circles using a graphing utility. The provided equations in the system are:\[x^{2}+y^{2} =25 \ (x-8)^{2}+y^{2} =41\]The plot depicts two circles where the equation of the smaller circle on the left is \(x^{2}+y^{2} =25\) and the larger one on the right is \((x-8)^{2}+y^{2} =41\). Initially, it can be observed that these circles intersect at two points.

Step 2: Mathematical Approximation

From the graphs, it can be roughly estimated that the points of intersection are near (4, 3) and (4, -3). These estimations can then be confirmed and refined by algebraic means. For this purpose, subtract one equation from the other to make x the subject. Then insert this expression into the other equation and solve the ensuing quadratic equation for y to find the exact coordinates for y.

Step 3: Verification

To verify these solutions, insert the calculated x and y values into both original equations to see if they fit. The correct solutions should make both of the equations in the system true.

Key Concepts

Graphing Utility

The marvelous digital tool known as a graphing utility helps visualize complex relationships between variables in mathematical equations. By inputting an equation, such as \(x^2 + y^2 = 25\) or \( (x-8)^2 + y^2 = 41\), the utility swiftly plots the respective shapes, in this case, two circles, onto a graph. This visual representation is especially invaluable when dealing with quadratic equations or systems of equations, as it provides an immediate picture of how these equations interact with each other—namely, where their graphs intersect. Utilizing graphing utilities facilitates a better conceptual understanding before delving into more intricate algebraic solutions.

Points of Intersection

When two graphs meet, their meeting points are called points of intersection. In algebraic terms, these points satisfy all equations involved. Visually identifying these points is often the first step in solving a system of equations graphically. After plotting the equations using a graphing utility, look for the points where the graphs coincide. For instance, observing the circles \(x^2 + y^2 = 25\) and \( (x-8)^2 + y^2 = 41\) intersect suggests likely candidates for solutions. These points are crucial as they are the ‘solutions’ to the system of equations, and can then be verified algebraically for exact values.

Quadratic Equations

Quadratic equations, which can be recognized by their characteristic \(ax^2 + bx + c = 0\) form, often graph as parabolas. However, when they come in pairs and both include squared terms of both variables (\