Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, use any method to solve the system of nonlinear equations. $$ \begin{array}{l} x^{4}-x^{2}=y \\ x^{2}+y=0 \end{array} $$

Step-by-Step Solution

Answer

The solution is \( (x, y) = (0, 0) \).

Step 1: Analyze the System of Equations

We have two equations: 1. \( x^4 - x^2 = y \) and 2. \( x^2 + y = 0 \). Notice that the second equation can be solved for \( y \): \( y = -x^2 \). This simplifies our task by expressing \( y \) in terms of \( x \).

Step 2: Substitute and Simplify

Substitute the expression for \( y \) from the second equation into the first equation:\[ x^4 - x^2 = -x^2 \]This simplifies to:\[ x^4 = 0 \].

Step 3: Solve the Simplified Equation

Solve the equation \( x^4 = 0 \). The only real solution is \( x = 0 \).

Step 4: Determine Corresponding y Value

Using the value \( x = 0 \), substitute back into \( y = -x^2 \). This gives us \( y = -0^2 = 0 \).

Step 5: Verify the Solution

Check that the values \( x = 0 \) and \( y = 0 \) satisfy both equations: 1. Equation 1: \( x^4 - x^2 = 0 - 0 = 0 \) which equals \( y = 0 \). 2. Equation 2: \( x^2 + y = 0 + 0 = 0 \), which holds true.Both equations are satisfied, confirming the solution.

Key Concepts

System of Equations

A system of equations is a set of two or more equations with multiple variables that are solved together. In our example, we encounter two equations:

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

In these equations, the variables \(x\) and \(y\) have interdependent relationships, meaning the value of one variable can affect the other. The goal is to find values for both \(x\) and \(y\) that satisfy each equation at the same time. It can be challenging when these equations are nonlinear, such as with exponents greater than one, but with systematic methods, we can solve for \(x\) and \(y\) to find a solution.

Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves isolating one variable in one of the equations and then substituting that expression into another equation. In our exercise, we start by expressing \(y\) in terms of \(x\) from the second equation:

• \(y = -x^2\)

This effectively reduces the number of variables in one equation.
By substituting \(-x^2\) for \(y\) in the first equation, we simplify it to a single-variable equation:

• \(x^4 - x^2 = -x^2\)

Thus, the substitution method helps make the problem easier to work with by reducing complexity.

Algebraic Solution Methods

Algebraic solution methods are the mathematical procedures applied for finding the values of unknowns in equations. In algebra, techniques such as factoring, expanding, and combining like terms are often used to simplify equations.
In our problem, once we've substituted the expression for \(y\), the equation

• \(x^4 - x^2 = -x^2\)

can be further simplified by adding \(x^2\) to both sides:

• \(x^4 = 0\)

This equation can then be solved using standard algebraic methods to find \(x\). In this case, because \(x^4 = 0\) yields only one solution

• \(x = 0\)

By undertaking each algebraic step carefully, we ensure that our solution satisfies the original conditions.

Solving Quadratic Equations

While our main example in the problem didn't specifically require solving a classical quadratic equation, understanding quadratic principles is beneficial. A quadratic equation typically appears in the form \(ax^2 + bx + c = 0\) and solving these can involve several approaches:

• Factoring, when possible
• Using the quadratic formula
• Completing the square

In our problem, the equivalent quadratic form is hidden within the substitution method step. Once simplified, \(x^2 + 0 = 0\) had already implied a quadratic form, solving naturally with \(x=0\) as the root.
Thus, knowledge of quadratics is often foundational even in seemingly simple solutions. For nonlinear equations, spotting these forms can save time and ensure accuracy.