Problem 27 - Algebra and Trigonometry

Question

\(19-32\) . Find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{x^{2}+y^{2}=9} \\ {x^{2}-y^{2}=1}\end{array}\right. $$

Step-by-Step Solution

Answer

The solutions are \((\sqrt{5}, 2), (\sqrt{5}, -2), (-\sqrt{5}, 2), (-\sqrt{5}, -2)\).

Step 1: Understanding the Problem

We are given a system of two equations: \( x^2 + y^2 = 9 \) and \( x^2 - y^2 = 1 \). Our objective is to find values of \( x \) and \( y \) that satisfy both equations simultaneously.

Step 2: Solving for \( x^2 \) and \( y^2 \)

Add the two equations: \( (x^2 + y^2) + (x^2 - y^2) = 9 + 1 \). Simplifying this gives \( 2x^2 = 10 \), so \( x^2 = 5 \). Subtract the second equation from the first: \( (x^2 + y^2) - (x^2 - y^2) = 9 - 1 \). Simplifying this gives \( 2y^2 = 8 \), so \( y^2 = 4 \).

Step 3: Finding \( x \) and \( y \)

Since \( x^2 = 5 \), \( x = \sqrt{5} \) or \( x = -\sqrt{5} \). Since \( y^2 = 4 \), \( y = 2 \) or \( y = -2 \).

Step 4: Verifying Solutions

We need to check which combination of \( x \) and \( y \) values satisfy both original equations. Substituting each combination:1. \((x, y) = (\sqrt{5}, 2)\): - \( x^2 + y^2 = 5 + 4 = 9 \) - \( x^2 - y^2 = 5 - 4 = 1 \) - Both equations are satisfied.2. \((x, y) = (\sqrt{5}, -2)\): - \( x^2 + y^2 = 5 + 4 = 9 \) - \( x^2 - y^2 = 5 - 4 = 1 \) - Both equations are satisfied.3. \((x, y) = (-\sqrt{5}, 2)\): - \( x^2 + y^2 = 5 + 4 = 9 \) - \( x^2 - y^2 = 5 - 4 = 1 \) - Both equations are satisfied.4. \((x, y) = (-\sqrt{5}, -2)\): - \( x^2 + y^2 = 5 + 4 = 9 \) - \( x^2 - y^2 = 5 - 4 = 1 \) - Both equations are satisfied.

Key Concepts

Quadratic Equations

Quadratic equations are an essential part of algebra and appear in various forms. In the context of our system of equations, we encounter quadratic relationships between variables. A quadratic equation is typically represented as \( ax^2 + bx + c = 0 \). However, in our system, the equations are in the form of \( x^2 + y^2 = 9 \) and \( x^2 - y^2 = 1 \). Here, the 'quadratic nature' expresses relationships where both the \( x \) and \( y \) variables are squared.

This characteristic means that each variable can have two potential solutions. These are \( x = \sqrt{5} \) or \( x = -\sqrt{5} \) for \( x^2 = 5 \) and \( y = 2 \) or \( y = -2 \) for \( y^2 = 4 \). This dual solution concept stems from the squared terms, allowing each variable to take on both positive and negative values while still satisfying the equation.

Solution Verification

Solution verification ensures that proposed solutions satisfy the original system of equations. Once we have potential values for \( x \) and \( y \), it's crucial to check if they fit both equations in the system. Verification involves substituting these values back into both equations.

Consider our solutions:

• \((x, y) = (\sqrt{5}, 2)\)
• \((x, y) = (\sqrt{5}, -2)\)
• \((x, y) = (-\sqrt{5}, 2)\)
• \((x, y) = (-\sqrt{5}, -2)\)

Each of these must be checked against \( x^2 + y^2 = 9 \) and \( x^2 - y^2 = 1 \). When substituted, all combinations yield values that correctly satisfy both equations.

This process confirms that our solutions are accurate. It's an important step because any oversight or incorrect calculations could lead to false answers, impacting the reliability of the solution. Hence, always ensure each solution meets the required conditions of the system.

Algebraic Manipulation

Algebraic manipulation is a method used to simplify and solve equations. In our exercise, it involves combining and rearranging equations to isolate specific variables or terms.

For instance, adding the two given equations \( x^2 + y^2 = 9 \) and \( x^2 - y^2 = 1 \) helps us find \( x^2 \). The use of addition and subtraction here simplifies the equations into \( 2x^2 = 10 \), leading to \( x^2 = 5 \). Similarly, subtracting one equation from the other allows us to isolate \( y^2 \) as \( 2y^2 = 8\), which gives \( y^2 = 4 \).

These operations are a critical part of solving systems of equations. They leverage the properties of equality and operations to manipulate and find the desired solutions efficiently. Understanding algebraic manipulation equips you with the tools to solve complex problems with ease.