Problem 75 - College Algebra with Modeling and Visualization

Question

Solve the system of equations. $$ \begin{aligned} (x-2)^{2}+y^{2} &=9 \\ x^{2}+y^{2} &=9 \end{aligned} $$

Step-by-Step Solution

Answer

The solutions are \((1, 2\sqrt{2})\) and \((1, -2\sqrt{2})\).

Step 1: Equation Simplification

Let's start by simplifying the given equations. We have two equations: 1. \((x-2)^2 + y^2 = 9\) 2. \(x^2 + y^2 = 9\). Notice that both right sides equal 9 and both contain \(y^2\).

Step 2: Subtract Equations

Subtract the second equation from the first to eliminate \(y^2\). \((x-2)^2 + y^2 - (x^2 + y^2) = 9 - 9\) This simplifies to: \((x-2)^2 - x^2 = 0\).

Step 3: Simplify Resulting Equation

Expand \((x-2)^2\) to simplify the expression: \(x^2 - 4x + 4 - x^2 = 0\) This reduces to: \(-4x + 4 = 0\).

Step 4: Solve for x

Solving \(-4x + 4 = 0\) gives: \(-4x = -4\) \(x = 1\).

Step 5: Substitute x into Second Equation

Now, substitute \(x = 1\) back into the second equation \(x^2 + y^2 = 9\): \(1^2 + y^2 = 9\) \(1 + y^2 = 9\).

Step 6: Solve for y

Solve for \(y^2\) by subtracting 1: \(y^2 = 8\) Taking the square root gives \(y = \pm \sqrt{8} = \pm 2\sqrt{2}\).

Step 7: Identify Solution Pair

The solution to the system of equations is: \((x, y) = (1, 2\sqrt{2})\) or \((x, y) = (1, -2\sqrt{2})\).

Key Concepts

Equation Solving

Solving a system of equations involves finding values for the variables that satisfy all the equations simultaneously. In this particular case, we began with two equations which share a common term, \(y^2\). This similarity allowed us to simplify our approach to discover the values of \(x\) and \(y\) that fit both equations.

• We used subtraction to eliminate one of the variables, \(y^2\). This is considered a strategic move in simplifying systems.
• The elimination resulted in an equation that only involved \(x\), making it straightforward to solve for \(x\).
• Once \(x\) was found, back substitution into one of the original equations yielded the solution for \(y\).

This process of solving a system often requires identifying shared components between equations. Simplifying each part separately can make the entire system fall into place more easily.

Algebra

Algebra is a powerful mathematical tool used to manipulate equations and expressions to find unknown variables. In the exercise, several algebraic techniques were employed:

• Equation expansion: We expanded \((x-2)^2\) to \(x^2 - 4x + 4\) to simplify the problem.
• Like-term cancellation: By subtracting entire equations, similar terms cancel, reducing complexity.
• Transposition and solving: The manipulation of equations to isolate variables, as shown when solving \(-4x + 4 = 0\).

Throughout algebra, it's essential to maintain equality by performing equal operations on both sides of an equation. This statement assures that the original problem is neither distorted nor simplified inaccurately.

Mathematical Modeling

Mathematical modeling involves representing real-world situations using mathematical concepts and equations. It's like building a bridge between math and the scenarios we encounter in reality.

• In our exercise, the equations could model a geometric shape, particularly circles. The expressions in the equations suggest constraints that the solutions must satisfy, forming a model.
• Finding the intersection point (solution) means identifying where the conditions described by each equation exist simultaneously, much like finding where two circles might intersect.
• Modeling often simplifies decision-making or prediction by allowing complex real-world interactions to be analyzed mathematically.

This modeling provides insight into how different variables interact and evolve, offering valuable predictions and understanding.