Problem 26 - College Algebra

Question

For the following exercises, use any method to solve the nonlinear system. $$\begin{aligned} x^{2}-y^{2} &=9 \\ y &=3 \end{aligned}$$

Step-by-Step Solution

Answer

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

Step 1: Substitute known value for y

Since the equation \( y = 3 \) is given, we substitute this value into the first equation \( x^2 - y^2 = 9 \).

Step 2: Perform the substitution

Substitute \( y = 3 \) into the first equation: \( x^2 - 3^2 = 9 \).

Step 3: Simplify the expression

Calculate \( 3^2 = 9 \), which transforms the equation to \( x^2 - 9 = 9 \).

Step 4: Solve for x

Add 9 to both sides to obtain \( x^2 = 18 \). Then take the square root of both sides to solve for \( x \): \( x = \pm \sqrt{18} \).

Step 5: Simplify the square root

Simplifying \( \sqrt{18} \), we have \( \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \).

Step 6: Write the solution

The solutions for the system are \( x = 3\sqrt{2} \) and \( x = -3\sqrt{2} \), with \( y = 3 \).

Key Concepts

Solving Equations

Solving equations involves finding values for variables that make an equation true. When dealing with a nonlinear system of equations, like the one in the exercise, different types of functions might be involved. For example, in the system given \( x^2 - y^2 = 9 \) and \( y = 3 \), the first equation involves squares of the variables, which makes it nonlinear.
To solve such systems, you can use various methods, such as substitution, elimination, or graphing. The goal is to find a solution where all equations are satisfied simultaneously. This often involves algebraic manipulation to isolate and solve for each variable.

Algebraic Substitution

Algebraic substitution is a powerful technique for solving systems of equations. It involves replacing one variable with an expression obtained from another equation. This allows you to reduce the number of variables, making the problem easier to solve. In our exercise, the equation \( y = 3 \) provides a straightforward way to substitute for \( y \) in the first equation.
By substituting \( y = 3 \) into \( x^2 - y^2 = 9 \), we simplify the equation to \( x^2 - 3^2 = 9 \). This step reduces the complexity of the system and allows us to focus on solving for \( x \).

• Substitution simplifies the original problem.
• Focus on substituting expressions that simplify the equation most.

Square Roots

Square roots are an important concept frequently encountered in solving equations, particularly those that involve quadratics or other power functions. When solving an equation like \( x^2 = 18 \), taking the square root of both sides helps to find the value of \( x \). It is important to remember that square roots have both a positive and a negative solution.
In our example, when we take the square root of 18, we find two solutions: \( x = 3\sqrt{2} \) and \( x = -3\sqrt{2} \).

• Always consider both positive and negative roots in equations dealing with squared terms.
• Simplifying square roots can involve breaking them down into smaller factors.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. Solving equations often involves simplifying these expressions. For instance, to simplify \( \sqrt{18} \), we express it as \( \sqrt{9 \times 2} \), which breaks it down further to \( 3\sqrt{2} \). Such simplifications can help when dealing with complex equations by making calculations more manageable.
Understanding the structure of algebraic expressions is crucial to simplify them effectively.

• Look for common factors to reduce expressions.
• Combine like terms where possible.