Problem 66 - College Algebra

Question

Explain how to solve a nonlinear system using the addition method. Use \(x^{2}-y^{2}-5\) and \(3 x^{2}-2 y^{2}-19\) to illustrate your explanation.

Step-by-Step Solution

Answer

The solutions of the system are (-3,-1) and (3,1)

Step 1: Write Down the Equations

The equations are \(x^{2}-y^{2}-5=0\) and \(3x^{2}-2y^{2}-19=0\)

Step 2: Add the equations

Add the two equations together. This process eliminates the \(y^{2}\) term since \(-y^{2} + (-2y^{2}) = -3y^{2}\), and the total sum becomes: \(x^{2} + 3x^{2} -3y^{2} -24 = 0\), or \(4x^{2}-3y^{2}-24=0\)

Step 3: Solve for one variable

For simplicity, solve the equation for \(y^{2}\). Rearrange the equation to isolate \(y^{2}\) on the left side of the equation: \(3y^{2}=4x^{2} -24\), or \(y^{2}=\frac{4x^{2}-24}{3}\)

Step 4: Substitute the value of \(y^{2}\) back into the original equation

Substitute the value of \(y^{2}\) that was found in step 3 into the first original equation: \(x^{2}-(\frac{4x^{2}-24}{3})-5=0\). Multiply everything by 3 to get rid of the fraction: \(3x^{2}-4x^{2}+24 -15=0\), and simplify to get \(-x^{2}+9=0\) or \(x^{2}=9\)

Step 5: Solve for the remaining variable

Taking the square root of both sides of the equation \(x^{2}=9\), gives two possible solutions for x, \( x=-3\) and \( x=3\), substitute each value for x into the equation found in step 3 ( \(y^{2}=\frac{4x^{2}-24}{3}\) ) to find the corresponding values of y. This gives the two solutions as (-3,-1) and (3,1).

Key Concepts

Addition Method

The addition method, also known as the elimination method, is an efficient technique used to solve systems of equations. The goal is to eliminate one of the variables by adding or subtracting the equations. This helps reduce the system to a simpler, more manageable equation with a single variable.

For this purpose, it's essential for the terms involving one of the variables to cancel out. By carefully adding or subtracting the equations, you can strategically eliminate one of the variables present in the system.

• Start by aligning the equations vertically, ensuring all like terms are in the same column.
• The coefficients of the variable you wish to eliminate should be the sum of zero when the equations are added.
• Once a variable is eliminated, solve the equation for the remaining variable.

This method works well for both linear and nonlinear systems, as illustrated in the example exercise. Here, we eliminated the variable \( y^2 \) to simplify the system.

Algebraic Equations

Algebraic equations are mathematical statements that contain one or more unknowns represented as variables. They form the basis for solving various types of problems, including those with nonlinear systems.

An equation like \(x^2-y^2-5=0\) involves powers and products of variables, classifying it as a nonlinear equation. Such equations require careful manipulation, often involving techniques such as factoring or setting terms equal to isolate variables.

Algebraic equations can express relationships between unknowns and constants, allowing us to explore solutions that fit these relationships. Whether dealing with simple linear forms or complex quadratic variations, understanding the structure of these equations is fundamental in both finding and interpreting solutions.

• Identify each variable and its relationship to others in the equation.
• Apply rules of algebra like distribution, combination of like terms, and isolation of variables.
• Manage complex expressions by simplifying them step by step.

Grasping these concepts enables confident solving of nonlinear systems using methods like substitution and elimination.

Solving Equations

Solving equations involves finding the values of variables that satisfy a given equation. In the case of nonlinear systems, this process can be intricate due to the degrees of variables and their interactions.

The process begins by setting up the equations, aligning and preparing them for manipulation using techniques such as the addition method. Once one of the variables is eliminated, the next step is to solve for the remaining variable.

For example, in our exercise, after eliminating \( y^2 \), we solved the resulting equation for \( x^2 \). The goal here is to reduce complex expressions to a solvable form, where substitution or isolation of the variable becomes possible.

• Isolate one variable to express it in terms of others which helps in substitution.
• Perform arithmetic operations to simplify and solve the equation.
• Cross-check solutions by substituting back into the original equations.

This methodology ensures all potential solutions are captured, and any errors in manipulation are identified and corrected effectively.

Square Roots

The square root operation is a critical tool in solving quadratic relationships, especially within systems of nonlinear equations. It specifically allows us to determine the actual numerical solutions of variables once they are expressed in squared forms.

In the context of our example, once we isolated \( x^2 = 9 \), we used square roots to find the values for \( x \). The square root of a number \( n \) is a value \( x \) such that \( x^2 = n \). Keep in mind, when taking the square root of a variable, there are two possible results: one positive and one negative.

For instance, \( x^2 = 9 \) implies that \( x = 3 \) or \( x = -3 \), reflecting the nature of quadratic equations.

• Always consider both positive and negative roots when solving \( x^2 = n \).
• Verify solutions by plugging them back into the original equation.
• Ensure all procedural steps are clear to track both numerical possibilities.

Understanding how square roots interact with variables allows comprehensive solving of these nonlinear systems, paving the way for more reliable solutions.