Problem 10 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The system of equations has no real solutions.

Step 1: Use the First Equation

Start by examining the first equation, \( x = 2 \). You can directly substitute \( x \) with 2 in the second equation since \( x \) has been solved for.

Step 2: Substitute into Second Equation

Substitute \( x = 2 \) into the second equation, which is \( x^2 - y^2 = 9 \). This gives you \( 2^2 - y^2 = 9 \).

Step 3: Solve for \( y^2 \)

Calculate \( 2^2 \) to get 4. The equation becomes \( 4 - y^2 = 9 \). Rearrange to find \( y^2 \): \( y^2 = 4 - 9 \).

Step 4: Simplify the Expression

Simplify \( y^2 = 4 - 9 \) to get \( y^2 = -5 \). This results in a negative number under the square root, indicating no real solutions for \( y \).

Step 5: Conclusion

Since \( y^2 = -5 \) has no real solutions (as the square root of a negative number is not real), the system of equations has no real solutions.

Key Concepts

Understanding a System of Equations

A System of Equations is when you have two or more equations working together. Each equation shares the same unknowns, like "x" or "y." The idea is to find one value for each variable that makes all the equations true at the same time. It’s a bit like a puzzle where you need to fit all pieces perfectly together.
In our exercise, the system consists of two equations:

• First, we have the equation: \( x = 2 \)
• Second, the nonlinear equation: \( x^2 - y^2 = 9 \)

Solving these will give us the values of "x" and "y" that satisfy both conditions. Nonlinear equations like the second one can make this more challenging, as they don't just involve simple lines when you graph them.

Using the Substitution Method

The substitution method means replacing a variable with its known value. First, you solve one of the equations to express one variable in terms of the others. Then, you take that expression and substitute it in the other equation.
For the problem given:

• We already know from the first equation that \( x = 2 \).
• So, we use this value of "x" to substitute in the second equation.

When you plug \( x = 2 \) into \( x^2 - y^2 = 9 \), it simplifies to \( 2^2 - y^2 = 9 \). This approach helps in breaking down complex problems into simpler parts. However, remember, substitution is powerful only when you solve one equation in terms of a single variable first.

Why Are There No Real Solutions?

Finding Real Solutions means finding numbers that we can see and understand directly, without involving any imaginary numbers. We typically consider solutions for variables on the real number line.
During the solving steps:

• We simplified our equation to \( y^2 = -5 \).
• This presents a snag: there’s no real number you can square to get a negative number. Squaring any real number, whether positive or negative, results in a positive value!

Hence, in this situation, \( y^2 = -5 \) offers no real solutions for "y." If all parts of a system don’t result in real solutions, the system itself lacks real solutions as well.

Navigating Algebra Problems

Algebra Problems often involve finding unknowns, referred to as variables. You need to manipulate equations using algebraic rules to discover these unknowns. This particular problem taught us that observation and examination are crucial.
Key steps when dealing with algebra problems:

• Analyze the given equations: What are you solving for?
• Pick a strategy: Here, we used substitution.
• Calculate carefully: Ensure calculations are correct, especially when it involves signs (positive/negative).

While some solutions might not exist in the real world, as seen in our system, the attempt and methodical approach to solving these problems enrich your understanding of equations and their behaviors.