Problem 22 - College Algebra

Question

Use any method to solve the system of nonlinear equations. $$ \begin{array}{r} x^{4}-x^{2}=y \\ x^{2}+y=0 \end{array} $$

Step-by-Step Solution

Answer

The solution to the system is \((0, 0)\).

Step 1: Substitute the Second Equation into the First

We have the system of equations: 1. \( x^4 - x^2 = y \)2. \( x^2 + y = 0 \)Start by substituting the value of \( y \) from the second equation into the first. The second equation gives us: \[ y = -x^2 \]Substitute \( y = -x^2 \) into the first equation:\[ x^4 - x^2 = -x^2 \]

Step 2: Simplify the Equation

Simplify the equation obtained after substitution:\[ x^4 - x^2 = -x^2 \]Add \( x^2 \) to both sides to simplify:\[ x^4 = 0 \]

Step 3: Solve for x

Now solve the equation \( x^4 = 0 \). Since any real number raised to the power of 4 equals 0 only when the base is 0, we have:\[ x = 0 \]

Step 4: Solve for y Using x

Substitute the value of \( x = 0 \) back into the equation for \( y \):Since \( y = -x^2 \):\[ y = -(0)^2 \]So, \( y = 0 \).

Step 5: Write the Solution as Ordered Pair

Combine the solutions for \( x \) and \( y \) into an ordered pair. The solution to the system of nonlinear equations is:\[ (x, y) = (0, 0) \]

Key Concepts

System of Equations

A system of equations consists of multiple equations that are solved together. Each equation may have two or more variables, and finding a common solution means identifying values for these variables that satisfy every equation in the set.

In this exercise, the system consists of two nonlinear equations:

• \( x^4 - x^2 = y \)
• \( x^2 + y = 0 \)

The goal is to find values of \( x \) and \( y \) that make both equations true at the same time. Solving systems of equations like this one often requires techniques beyond simple arithmetic because they may involve nonlinear relationships.

The presence of nonlinear terms, such as \( x^4 \) and \( x^2 \), means that the equations cannot be graphed as straight lines on a coordinate plane. Instead, these terms create curves, making the solving process intriguing but somewhat more complex.

Understanding how these equations represent certain geometric shapes can deepen your comprehension of why specific methods are applied to solve them.

Substitution Method

The substitution method is a technique for solving systems of equations where you solve one of the equations for one variable and then substitute this expression into the other equation. This method simplifies the problem by reducing the number of variables in the equations.

In the given system, we used the second equation \( x^2 + y = 0 \) to express \( y \) in terms of \( x \):

• \( y = -x^2 \)

Substituting \( y = -x^2 \) into the first equation \( x^4 - x^2 = y \), transforms it into a single-variable equation:

• \( x^4 - x^2 = -x^2 \)

Through substitution, we've effectively removed one variable, simplifying the solution process. This step is crucial because it allows us to focus on one variable at a time while ensuring that any solution satisfies both equations in the original system.

This method is particularly effective when one of the equations is easily solvable for one variable, which can then be substituted into the other equations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make finding a solution easier. It often requires skills such as expanding, factoring, or canceling terms, and is integral to the problem-solving process.

In solving the current system of equations, once substitution is used, we simplify the equation \( x^4 - x^2 = -x^2 \) to \( x^4 = 0 \).

• Add \( x^2 \) to both sides: \( x^4 = 0 \)

From here, deducing \( x \) is straightforward since the equation reads as any number raised to the fourth power equaling zero.

Such manipulation allows us to focus on an equation that can be easily solved for \( x \), confirming that the only real number satisfying \( x^4 = 0 \) is \( x = 0 \). By performing algebraic manipulation competently, we significantly streamline the path to finding the values of \( x \) and ultimately \( y \).

Ordered Pair Solution

An ordered pair solution represents a solution to a system of equations, expressing the values of the variables as a pair \( (x, y) \). This pair indicates a specific point on a coordinate plane where both equations are satisfied simultaneously.

In this exercise, once we determined that \( x = 0 \) through solving the simplified equation, we substituted back to find \( y \):

• \( y = -x^2 = 0 \)

Thus, \( (x, y) = (0, 0) \) becomes our ordered pair solution, indicating the point where the curves represented by the original nonlinear equations intersect.

Finding the ordered pair not only solves the problem but also provides a geometrical interpretation of the solution. It marks the exact convergence point of the two equations, demonstrating concretely how the variables relate to each other within the conceptual framework of the system.