Problem 26 - College Algebra

Question

Use any method to solve the nonlinear system. $$ \begin{array}{r} x^{2}-y^{2}=9 \\ y=3 \end{array} $$

Step-by-Step Solution

Answer

Solutions are \((3\sqrt{2}, 3)\) and \((-3\sqrt{2}, 3)\).

Step 1: Substitute the value of y

We are given the equation \( y = 3 \). Substitute this value into the first equation of the system, \( x^2 - y^2 = 9 \). This gives you: \( x^2 - 3^2 = 9 \).

Step 2: Simplify the equation

Simplify the equation obtained from Step 1: \( x^2 - 9 = 9 \). Add 9 to both sides to get \( x^2 = 18 \).

Step 3: Solve for x

Find the value of \( x \) by taking the square root of both sides of the equation \( x^2 = 18 \). This results in \( x = \sqrt{18} \) or \( x = -\sqrt{18} \). Simplify \( \sqrt{18} \) to get \( x = 3\sqrt{2} \) and \( x = -3\sqrt{2} \).

Step 4: Present the solutions

Since \( y = 3 \), and we have values for \( x \), the solutions for the system are \( (x, y) = (3\sqrt{2}, 3) \) and \( (x, y) = (-3\sqrt{2}, 3) \).

Key Concepts

Substitution Method

When tackling systems of equations, one useful strategy is the substitution method. In our exercise, this technique acts as a powerful tool because it allows us to deal with simpler expressions, making the problem more manageable.
The substitution method involves replacing one variable with an expression derived from another equation in the system. Here's how it works in our example:

• We start with two equations:
  • \(x^2 - y^2 = 9\)
  • \(y = 3\)
• Given that \(y = 3\), replace all occurrences of \(y\) in the first equation, transforming it into \(x^2 - 3^2 = 9\).

The key here is the reduction of complexity from dealing with two variables to focusing on just one.
Substitution sets the stage for further simplification and progresses us right into solving for the unknown.

Solving Algebraic Equations

Once substitution is used, we end up with a simplified algebraic equation that needs solving: \(x^2 - 9 = 9\).
The goal is to isolate the unknown, \(x\), and find its value.

• First, adjust the equation by balancing both sides. Add 9 to each side to eliminate -9, resulting in \(x^2 = 18\).
• Now, \(x^2\) stands alone, and we're equipped to solve for \(x\).

At this point, the equation is in its simplest form, paving the way for finding the solution through square root extraction.
The process is not just about arithmetic—it's also about logical deduction and verifying each transformation to ensure we're on track.

Square Root Simplification

With \(x^2 = 18\) isolated, the next step involves calculating the square root.
Finding square roots enables us to solve for \(x\), since squaring and square roots are inverse operations.

• By taking the square root of both sides, we derive \(x = \sqrt{18}\) and \(x = -\sqrt{18}\). This step considers both positive and negative roots because squaring negates sign changes.
• Simplifying \(\sqrt{18}\) involves recognizing it as \(3\sqrt{2}\). This is done by factoring 18 into \(9 \times 2\), where \(\sqrt{9}\) simplifies to 3.

This simplification is critical as it refines our solution into its simplest form, providing clarity on the actual values of \(x\).
Pay attention to straightforward methods like these to enhance accuracy and understanding.

System of Equations

A system of equations consists of multiple equations that are solved simultaneously. In our nonlinear system, we have:

• An equation involving \(x^2\) and \(y^2\), \(x^2 - y^2 = 9\), which is nonlinear due to the squared terms.
• A linear equation, \(y = 3\), which directly provides the value for \(y\).

The main aim is to find values for both \(x\) and \(y\) that satisfy all the equations involved.
Systems like these often require combining different strategies—here, substitution was key to reducing complexity.
After processing each equation, our solution becomes: \((x, y) = (3\sqrt{2}, 3)\) and \((x, y) = (-3\sqrt{2}, 3)\). These solutions demonstrate how the given \(y\)-value integrates with possible \(x\)-values to answer the system effectively.
Understanding this interaction sharpens your ability to handle both linear and nonlinear systems with confidence.