Problem 14 - College Algebra

Question

Solve the system of nonlinear equations using elimination. $$ \begin{array}{r} y^{2}-x^{2}=9 \\ 3 x^{2}+2 y^{2}=8 \end{array} $$

Step-by-Step Solution

Answer

No real solution exists for this system.

Step 1: Recognize the System and Goal

The given system of equations is nonlinear and consists of two equations:\[ y^2 - x^2 = 9 \]\[ 3x^2 + 2y^2 = 8 \]Our goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously.

Step 2: Isolate Terms in One Equation

First, isolate a term from the first equation. Rewrite:\[ y^2 = x^2 + 9 \]This expression helps eliminate \(y^2\) by substitution into the second equation.

Step 3: Substitute into the Second Equation

Substitute \( y^2 = x^2 + 9 \) into the second equation:\[ 3x^2 + 2(x^2 + 9) = 8 \]Simplify to remove \(y^2\) from the second equation.

Step 4: Simplify and Solve for \(x^2\)

Expand and combine like terms to simplify the equation:\[ 3x^2 + 2x^2 + 18 = 8 \]Combine the \(x^2\) terms:\[ 5x^2 + 18 = 8 \]Subtract 18 from both sides:\[ 5x^2 = -10 \]Divide by 5:\[ x^2 = -2 \]Since \(x^2 = -2\) is not possible with real numbers, there is no real solution for \(x\).

Step 5: Analyze Result for Real Solutions

Since \(x^2 = -2\) has no real solutions, this suggests that there are no real values of \(x\) and \(y\) that satisfy both equations in the system. The solutions are complex numbers, outside the scope of this problem if we are considering only real solutions.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations that have a set of variables. When solving such systems, the goal is to find the values of each variable that satisfy all the equations simultaneously.
In this problem, the system of equations is nonlinear, meaning that the equations contain terms that are not simply the first power of each variable. The specific system given is:

• \( y^2 - x^2 = 9 \)
• \( 3x^2 + 2y^2 = 8 \)

For this system, both \(x\) and \(y\) need to be determined so that both equations hold true at the same time.

Elimination Method

The elimination method is a powerful technique used to solve systems of equations. It focuses on removing one variable so you can solve for the other. In this exercise, we first isolate terms in one equation to make substitution easier.
By isolating \( y^2 \), we had:

• From \( y^2 - x^2 = 9 \) to \( y^2 = x^2 + 9 \).

Then, we substituted \( y^2 = x^2 + 9 \) into the second equation to eliminate \( y^2 \) from that equation, simplifying the system to solve for \( x \). This allows us to focus on one variable at a time, making it easier to find solutions.

Real Solutions

Real solutions refer to solutions for the system where the variables take on real number values. In mathematical terms, real numbers include all the numbers on the number line, excluding imaginary numbers.
As we solved the given system, we found:

• After simplifying, we obtained \( 5x^2 = -10 \).
• Dividing by 5, the equation becomes \( x^2 = -2 \).

In real numbers, there are no solutions for \( x^2 = -2 \) because no real number squared will result in a negative. Thus, there are no real solutions for this system of equations.

Complex Solutions

When an equation solution is not found within the set of real numbers, we venture into complex numbers, which include an imaginary part. Complex solutions are of the form \( a + bi \), where \( i \) is the square root of \(-1\).
For this problem:

• We arrive at \( x^2 = -2 \), suggesting \( x = \pm \sqrt{-2} \).
• Using complex numbers, \( x = \pm i \sqrt{2} \).

If we assume similar for \( y \), complex solutions exist where variable values include imaginary parts. If complex solutions are outside your current scope, it’s enough to know they are needed when equations cannot be satisfied by real numbers alone. Complex solutions enable continued operations in mathematics where traditional real solutions assert limits.