Problem 27 - College Algebra

Question

Use any method to solve the nonlinear system. $$ \begin{array}{r} x^{2}-y^{2}=9 \\ x-y=0 \end{array} $$

Step-by-Step Solution

Answer

The system has no solutions due to the contradiction derived from the assumptions.

Step 1: Identify System of Equations

We are given a nonlinear system of equations: \( x^2 - y^2 = 9 \) and \( x - y = 0 \). The first equation is a difference of squares, while the second indicates that \( x \) and \( y \) are equal.

Step 2: Substitute One Variable for Another

From the equation \( x - y = 0 \), we deduce that \( x = y \). We will substitute \( x \) for \( y \) in the first equation.

Step 3: Rewrite First Equation

Substituting \( x = y \) into \( x^2 - y^2 = 9 \), we have \( x^2 - x^2 = 9 \), simplifying it to \( 0 = 9 \), which indicates a contradiction.

Step 4: Analyze System's Feasibility

The contradiction \( 0 = 9 \) suggests that there are no solutions to the system since the assumptions lead to an impossible result.

Key Concepts

difference of squares

The difference of squares is a mathematical concept that involves an expression of the form \( a^2 - b^2 \). This type of expression can be factored into two binomials: \((a+b)(a-b)\). In the given exercise, the equation \( x^2 - y^2 = 9 \) represents a difference of squares.
To solve problems involving the difference of squares, it's often helpful to recognize this pattern and factorize. For instance, in this equation, it can initially be written as \((x+y)(x-y)\), which is useful for understanding the relationship between the variables involved. Recognizing such patterns helps to utilize algebraic identities effectively for solving equations.

substitution method

The substitution method is a technique used to solve a system of equations where you solve one equation for one variable and then substitute that expression into the other equation. In this exercise, we use the substitution method by recognizing that the second equation, \( x-y=0 \), implies that \( x = y \).
This simplifies the system significantly. By substituting \( x = y \) into \( x^2 - y^2 = 9 \), we attempt to find a solution. All occurrences of \( y \) are replaced with \( x \), making it a single-variable problem. However, this leads to a contradiction in this particular problem, as we'll discuss further.

system of equations

A system of equations is a set of two or more equations with the same set of variables. In this exercise, we are dealing with a nonlinear system of equations because one of the equations involves terms that are not simply linear (straight lines on a graph) but has quadratic terms like \( x^2 \) and \( y^2 \).
Systems of equations are usually solved to determine the values of the variables that satisfy all the equations simultaneously. There are various methods to solve such systems, including substitution and elimination methods. In our case, the use of the substitution method directly highlights whether the system is consistent or not.

no solutions

An outcome of 'no solutions' in the context of a system of equations means that there is no set of values for the variables that can satisfy all equations involved. Here, after substituting \( x = y \) into \( x^2 - y^2 = 9 \), we ended up with a false statement: \( 0 = 9 \).
This contradiction shows that our assumption that \( x = y \) does not hold true, and therefore, no values for \( x \) and \( y \) exist that satisfy both equations simultaneously. Such results are important in understanding the nature of the problem and the relationships between variables. It's critical to recognize when a set of equations provides no valid solutions, indicating that the modeled situation or initial assumptions may be flawed or incompatible.