Problem 10 - College Algebra

Question

For the following exercises, solve the system of nonlinear equations using substitution. $$\begin{aligned} x &=2 \\ x^{2}-y^{2} &=9 \end{aligned}$$

Step-by-Step Solution

Answer

The system has no real solutions.

Step 1: Substitute the Value of x

We are given that \( x = 2 \). So, we will substitute \( x = 2 \) into the second equation \( x^2 - y^2 = 9 \). This gives us the equation \( 2^2 - y^2 = 9 \).

Step 2: Solve for y

From the equation \( 2^2 - y^2 = 9 \), simplify the left side to get \( 4 - y^2 = 9 \). To isolate \( y^2 \), subtract 4 from both sides, resulting in \( -y^2 = 5 \). Multiply both sides by -1 to obtain \( y^2 = -5 \).

Step 3: Analyze the Result

The equation \( y^2 = -5 \) implies there are no real solutions for \( y \) because a square of a real number cannot be negative. Therefore, the system does not have any real solutions.

Key Concepts

Substitution Method

The substitution method is a powerful technique used to solve systems of equations, including nonlinear equations. It involves substituting one equation into another in order to simplify the system and make it easier to solve. In the context of the given problem, the first equation is simple:

• \( x = 2 \)

You can directly substitute the value of \( x \) into the second equation:

• \( x^2 - y^2 = 9 \)

Substitute \( x = 2 \) gives us \( 2^2 - y^2 = 9 \). By replacing \( x \) with \( 2 \), this method reduces the system to a single equation in terms of \( y \). This makes it much simpler to solve than dealing with both variables simultaneously.

This technique is especially valuable when one of the equations is easily expressible, making it an efficient way to tackle such problems.

Solving Systems of Equations

Solving systems of equations involves finding a set of values for the variables that satisfy all equations within the system. These systems can be linear or nonlinear, and they may include two or more equations. In the provided exercise, we have a nonlinear system:

• \( x = 2 \)
• \( x^2 - y^2 = 9 \)

The goal is to find numbers for \( x \) and \( y \) that make both equations true. By employing the substitution method, we first determine \( x \) and use that information to simplify and solve the second equation. In this case, substituting \( x \) into the second equation appeared straightforward, but it led to complexity due to the nature of the resulting equation.

Such challenges are typical of nonlinear systems, where solutions sometimes don't exist in the set of real numbers. It's crucial to recognize not every system will yield a solution.

Real Solutions

Real solutions refer to solutions of equations or systems of equations where the solutions are real numbers. For our problem, after substituting \( x = 2 \), we ended up with the equation:

• \( y^2 = -5 \)

This is where a key concept arises. For any real number \( y \), \( y^2 \) can never be negative. The equation \( y^2 = -5 \) actually tells us there are no real numbers that satisfy the system. This shows that the system

• \( x = 2 \)
• \( x^2 - y^2 = 9 \)

does not have any real solutions.

Understanding when and why a system of equations has no real solutions is crucial, especially since it indicates the nature of the solutions, in this case informing that any possible solutions must lie in the complex number domain.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations using algebraic techniques. In solving the provided system, we used this form of manipulation to solve equations more effectively. Starting with:

• \( 2^2 - y^2 = 9 \)

We simplified the left-hand side to \( 4 - y^2 = 9 \), and then adjusted the equation to \( -y^2 = 5 \) by subtracting 4 from both sides. After multiplying by -1, we arrived at \( y^2 = -5 \).

These algebraic steps help isolate variables and consolidate them into solvable forms. A skilled use of algebraic manipulation allows one to identify inconsistencies or impossibilities in the solution set, as it did here by showing \( y^2 = -5 \), highlighting the absence of real solutions.