Problem 44 - Precalculus with Limits

Question

Solving a System of Equations Graphically In Exercises \(33-44,\) solve the system graphically. $$\left\\{\begin{aligned} x^{2}+y^{2} &=25 \\\\(x-8)^{2}+y^{2} &=41 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The short answer would be the coordinates of the intersecting points of the circles, which can be approximated from the graph. For an exact solution, algebraic or numerical methods would need to be used.

Step 1: Plot the First Circle

To solve this system graphically, start by plotting the first circle on a graph. The equation for this circle is \(x^2 + y^2 = 25\). This is a standard circle equation centered at the origin (0,0) with radius \(r = \sqrt{25} = 5\). Plot this circle by starting at the origin, measure out a distance of 5 units to the right, up, left, and down, and sketch the circle so it passes through these four points.

Step 2: Plot the Second Circle

Next, plot the second circle whose equation is \((x-8)^2 + y^2 = 41\). This equation represents a circle centered at (8,0) with a radius \(r=\sqrt{41}\) which is slightly larger than 6. To plot this circle, first locate the center at the point (8,0), then measure out a distance of around 6 units to the right, up, left, and down on your paper or graphing tool. Sketch the circle so it passes through these points.

Step 3: Identifying System Solution

The solution to the system of equations is the points where the circles intersect. After plotting, we will then identify the points where the two circles intersect. The interaction points are the x, y values that satisfy both equations simultaneously. These are the solutions to the system, and their approximate coordinates can be read from the graph.

Key Concepts

Graphing Solutions

Graphing solutions to systems of equations involves visually representing each equation on a coordinate plane. This makes it easier to see where the equations intersect, which is the solution to the system. When graphing a system that includes circles, as in the given exercise, you draw each circle based on its equation.

• First, identify the type of graph each equation represents. Here, both are perfect circles.
• Plot each circle using the information given about their centers and radii.
• Once both circles are plotted, observe their intersections — these will be the solutions.

Visual methods, like graphing, provide insight into the nature of solutions and can serve as a check against algebraic computations. This method is excellent for visually inclined learners and situations where rough solutions are acceptable or exploratory analysis is needed.

Circle Equations

Circle equations in mathematics are typically expressed in the standard form: \[ (x-h)^2 + (y-k)^2 = r^2 \]where \((h,k)\) represents the center of the circle, and \(r\) is the radius.
The first equation from the exercise, \(x^2 + y^2 = 25\), is a simpler form with the center at the origin \((0,0)\), and a radius of \(5\) units (since \(r^2 = 25\)). This represents a basic circle centered at the origin.
The second equation, \((x-8)^2 + y^2 = 41\), indicates a translation from the origin to \((8,0)\) and a radius around \(6.4\), meaning the circle is shifted right on the x-axis.
Understanding the structure of these equations is critical for correctly plotting the circles. Recognizing shifts and radius sizes ensures that the graph accurately represents the circles and, subsequently, the solutions.

Intersection Points

Intersection points are where the graphs of equations meet. For systems involving geometric shapes like circles, intersections occur where the shapes share common points.

• These points satisfy all equations in the system simultaneously.
• Graphically identifying these points involves observing where the plotted circles overlap.
• Accurate graphical depiction will show two intersection points for this particular exercise.

Once you establish the intersections on the graph, you can approximate these coordinates. These coordinates give you a visual representation of solutions and help verify algebraic solutions. In the context of real-world applications, understanding intersection points can represent common solutions in systems like networks, overlapping interest areas, or physical intersections in space. This understanding connects the geometry with practical analytical applications.