Problem 26 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Substitution

We are given two equations: \(x^2 - y^2 = 9\) and \(y = 3\). Notice that the second equation, \(y=3\), directly gives us the value of \(y\). We can substitute this value into the first equation to find \(x\).

Step 2: Substitute y in the First Equation

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

Step 3: Simplify the Equation

Perform the calculation \(3^2 = 9\), resulting in the equation \(x^2 - 9 = 9\).

Step 4: Solve for x

Add 9 to both sides of the equation to isolate \(x^2\): \(x^2 = 18\).

Step 5: Find Possible Values of x

Take the square root of both sides to find \(x\): \(x = \pm \sqrt{18}\). Simplify \(\sqrt{18}\) to \(3\sqrt{2}\). Therefore, \(x = 3\sqrt{2}\) or \(x = -3\sqrt{2}\).

Step 6: Final Step: State the Solution

The values of \(x\) are \(3\sqrt{2}\) and \(-3\sqrt{2}\) when \(y=3\). So the solutions to the system are \((3\sqrt{2}, 3)\) and \((-3\sqrt{2}, 3)\).

Key Concepts

Substitution Method

The substitution method is a common technique used to solve systems of equations. It involves solving one equation for a single variable and then substituting this value into the other equation. In the given exercise, the equation \( y = 3 \) provides a straightforward value for \( y \). By substituting \( y = 3 \) into the first equation \( x^2 - y^2 = 9 \), we eliminate \( y \) and simplify the system to only involve \( x \). This method is particularly useful in systems where one equation is already solved for a variable or can be easily rearranged to express one variable in terms of another.

• Find a variable with a simple or direct equation.
• Substitute this value into the other equation(s).
• Simplify and solve the resulting equation.

The substitution method helps reduce the complexity in nonlinear systems by breaking them down into a single equation with one unknown.

Solving Equations

After substituting the value of \( y \) into the first equation, we obtain \( x^2 - 9 = 9 \). Solving equations like this involves simplifying and rearranging terms to isolate the unknown variable.To solve for \( x \), the primary goal is to perform algebraic operations that will make the equation easier to interpret. First, the equation is simplified by moving constants to one side, which results in \( x^2 = 18 \). Breaking down the solving process can include:

• Performing arithmetic operations like addition or subtraction to simplify the equation.
• Applying inverse operations such as taking the square root to isolate the variable.

By focusing on these algebraic manipulations, we slowly unveil possible values for \( x \). For nonlinear systems, especially with square terms, solving often involves identifying when to apply roots or other inverse functions to handle equations efficiently.

Algebraic Solutions

Finding algebraic solutions lets us express values of variables in the simplest form possible. In our nonlinear system, solving for \( x \) resulted in \( x = \pm \sqrt{18} \), which simplifies further to \( x = \pm 3\sqrt{2} \). The use of square roots is typical in algebraic solutions, especially when dealing with quadratic terms. Simplifying \( \sqrt{18} \) to \( 3\sqrt{2} \) ensures the solution is precise and recognizably reduced.

• Taking the square root of both sides provides potential positive and negative values.
• Simplification helps represent solutions in the lowest terms.

Algebraic solutions give you the final values of your variables in a clear and organized manner. This step insists on verifying the correctness of simplified terms, ensuring all calculations align with the original equations. This technique is powerful for presenting and understanding solutions in a neat and exact way.