Problem 27 - College Algebra

Question

For the following exercises, use any method to solve the nonlinear system. $$\begin{aligned} x^{2}-y^{2} &=9 \\ x-y &=0 \end{aligned}$$

Step-by-Step Solution

Answer

There are no solutions to this system.

Step 1: Analyze and understand the equations

The system of equations we have is:1. \( x^2 - y^2 = 9 \) (Equation 1)2. \( x - y = 0 \) (Equation 2)The first equation is a nonlinear equation (a difference of squares), and the second equation is linear. We aim to find the values of \(x\) and \(y\) that satisfy both these equations.

Step 2: Solve the linear equation for one variable

From Equation 2, \(x-y=0\), we can solve for one of the variables. Adding \(y\) to both sides, we obtain:\[x = y\]This gives us a direct relationship between \(x\) and \(y\).

Step 3: Substitute into the first equation

Now, we use the expression from Step 2 (\(x = y\)) and substitute \(y\) with \(x\) in Equation 1:\[x^2 - x^2 = 9\]This substitution simplifies the equation.

Step 4: Simplify the equation

After substitution, the new equation is:\[0 = 9\]This is a contradiction, meaning our initial assumption or substitution should be reconsidered. However, it also implies there might be a mistake or special note needed for any re-evaluations.

Step 5: Re-evaluate the substitution

Re-evaluate if there could be another method or approach needed; however, when revisiting the condition \(x = y\), analyze any geometrical or conceptual mismatch on the functions. In standard checks, if one finds consistent zeros or dependent overlaps, recognize this as a potential empty solution set or conditional system based upon context.

Key Concepts

Difference of Squares

The term "difference of squares" refers to an expression of the form \( a^2 - b^2 \). It's a distinctive algebraic expression that can be easily factored using a special formula.
This formula is: \( a^2 - b^2 = (a + b)(a - b) \).
In the given exercise, the equation \( x^2 - y^2 = 9 \) is a difference of squares.

• Recognizing this pattern is crucial as it often simplifies the problem and provides a clearer path to find a solution.
• Factoring makes it easier to solve because it transforms the polynomial into a product of simpler expressions.

One of the main benefits of factoring equations like this is that it can expose hidden relationships between variables, or reveal solutions that might not be obvious at first glance.

Linear Equation

A linear equation is an equation involving variables to the first power, meaning there are no exponents other than one.
In this problem, the second equation \( x - y = 0 \) is linear.

• Such equations usually graph as straight lines.
• They can often be solved quickly through arithmetic operations like addition, subtraction, and substitution.

Linear equations form the foundation for understanding more complex, non-linear equations. In our exercise, solving for \( x \) in terms of \( y \) from the linear equation was a crucial step.
Understanding these simple relationships helps us to find or confirm any solutions when combined with other, potentially more complex conditions.

Substitution Method

The substitution method is a common technique used to solve systems of equations. It involves expressing one variable in terms of another and then substituting that expression into a different equation.
This method is particularly useful when one of the equations is easier to solve for a single variable.

• In our exercise, from \( x - y = 0 \), we find \( x = y \).
• We substitute this into the other equation \( x^2 - y^2 = 9 \).
• This substitution should simplify the entire process, as seen in the transformed equation.

With substitution, we potentially reduce a system with multiple variables into an equation with a single variable, making the problem much more straightforward. However, it's essential always to check the results as simplification might lead to contradictions or reveal special conditions.

Empty Solution Set

An empty solution set means there are no values for the variables that satisfy all the equations in the system.
In our example, when we substituted \( x = y \) into the equation \( x^2 - y^2 = 9 \), we ended up with the impossible equation \( 0 = 9 \).

• This outcome indicates a contradiction.
• Such situations typically suggest that the original equations have no points of intersection — there are no real values of \( x \) and \( y \) solving both equations simultaneously.
• In the context of our system, it implies that assuming \( x = y \) doesn't work within the structure of the given mathematical constraints.

Empty solution sets remind us that not every system of equations has a solution, and indicate the importance of carefully analyzing the entire system before proceeding.