Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system of nonlinear equations using elimination. $$ \begin{array}{l} x^{2}+y^{2}=25 \\ x^{2}-y^{2}=1 \end{array} $$

Step-by-Step Solution

Answer

Solutions: \((\sqrt{13}, 2\sqrt{3}), (\sqrt{13}, -2\sqrt{3}), (-\sqrt{13}, 2\sqrt{3}), (-\sqrt{13}, -2\sqrt{3})\)."

Step 1: Understanding the System

We have two nonlinear equations: \( x^2 + y^2 = 25 \) and \( x^2 - y^2 = 1 \). Our goal is to find the values of \( x \) and \( y \) that satisfy both equations.

Step 2: Setting Up Elimination

To eliminate one variable, we will add the two equations. By doing so, the \( y^2 \) terms will cancel out.

Step 3: Adding the Equations

Add the two equations:\[x^2 + y^2 + x^2 - y^2 = 25 + 1\]This simplifies to:\[2x^2 = 26\]

Step 4: Solving for x

To find \( x^2 \), divide both sides of the equation by 2:\[x^2 = \frac{26}{2} = 13\]Taking the square root, we find:\[x = \pm \sqrt{13}\]

Step 5: Substitute x back into an Equation

We substitute \( x^2 = 13 \) into the first equation \( x^2 + y^2 = 25 \):\[13 + y^2 = 25\]

Step 6: Solve for y

Subtract 13 from both sides:\[y^2 = 12\]Taking the square root, we find:\[y = \pm \sqrt{12}\] which further simplifies to \[y = \pm 2\sqrt{3}\]

Step 7: Obtain the Solutions

We found that \( x = \pm \sqrt{13} \) and \( y = \pm 2\sqrt{3} \), leading to the solution pairs: \((\sqrt{13}, 2\sqrt{3}), (\sqrt{13}, -2\sqrt{3}), (-\sqrt{13}, 2\sqrt{3})\), and \((-\sqrt{13}, -2\sqrt{3})\).

Key Concepts

Elimination Method

The elimination method is a powerful tool used to solve systems of equations. It focuses on removing one of the variables to make the solving process easier. This method involves either adding or subtracting the equations to cancel out one of the variables. By doing so, you reduce the two equations into one, enabling you to solve for one variable first.

• Start by aligning the equations to easily identify which variable you can eliminate.
• Add or subtract the equations depending on the need to cancel out a variable.
• Solve the resulting equation to find the value of one variable.
• Substitute this value back into one of the original equations to solve for the other variable.

In the exercise given, the elimination method effectively simplifies the equations by canceling out the \( y^2 \) terms through addition. This allows for straightforward calculation of the value of \( x \). Once \( x \) is found, you can easily find \( y \) by substitution.

System of Equations

A system of equations consists of two or more equations that are solved together. These equations share common variables, and solutions to the system satisfy all equations simultaneously. In practice, these systems can describe geometrical shapes, physical phenomena, or various scenarios in algebra.

• Each equation in the system must have the same set of variables.
• The solution is found when all equations are satisfied by the same values of those variables.

For the given exercise, the system of nonlinear equations includes \( x^2 + y^2 = 25 \) and \( x^2 - y^2 = 1 \). These equations are nonlinear because they involve quadratic terms. Solving this system provides values for \( x \) and \( y \) that solve both equations simultaneously. The challenge lies in managing these non-linear terms effectively, often by using methods like elimination.

Solving Equations Step-by-Step

Solving equations step-by-step is a logical approach to ensure each part of the solution process is understood and followed correctly. This structured method breaks down the problem into manageable pieces:

• Begin by clearly understanding each equation in the system. Identify what's being asked.
• Set up a plan using methods like elimination or substitution to solve for one of the variables.

In the provided solution:

• Add the equations to eliminate \( y^2 \), simplifying the system.
• Solve for \( x \) first, acquiring exact values using root calculations.
• Substitute the known \( x \) values back into one equation to solve for \( y \).
• Finally, derive all possible pairs of solutions that satisfy both original equations.

By following these steps, you derive a complete set of solutions for the nonlinear system. This systematic approach reduces errors and ensures that each solution pair is verified to satisfy all equations in the system.