Problem 67 - College Algebra with Modeling and Visualization

Question

If possible, solve the nonlinear system of equations. $$ \begin{aligned} &x^{3}-x=3 y\\\ &x-y=0 \end{aligned} $$

Step-by-Step Solution

Answer

The solutions are \((0,0)\), \((2,2)\), and \((-2,-2)\).

Step 1: Analyze the System of Equations

We have two equations in the system: \(x^3 - x = 3y\) and \(x - y = 0\). The second equation, \(x - y = 0\), can be rewritten as \(x = y\). This equation gives a direct relationship between \(x\) and \(y\).

Step 2: Substitute and Simplify

Substitute \(y = x\) from the second equation into the first equation: \(x^3 - x = 3x\). Simplifying this gives \(x^3 - x - 3x = 0\) which results in \(x^3 - 4x = 0\).

Step 3: Factor the Polynomial Equation

Factor the equation \(x^3 - 4x = 0\). This can be factored as \(x(x^2 - 4) = 0\). Further factor \(x^2 - 4\) using the difference of squares: \(x(x-2)(x+2) = 0\).

Step 4: Solve for the Roots

Solve the factored equation \(x(x-2)(x+2) = 0\) to find the roots. The solutions are \(x = 0\), \(x = 2\), and \(x = -2\).

Step 5: Find Corresponding y-values

Using the relationship \(y = x\), substitute the found \(x\) values into it to find \(y\). This gives the pairs \((x, y) = (0, 0)\), \((2, 2)\), and \((-2, -2)\).

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions that involve variables raised to whole-number exponents. These equations need to be set to equal zero to solve for the variable. In general, a polynomial equation can be written as:
\[ a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0 \]
where the coefficients \(a_n, a_{n-1}, \ldots, a_0\) are constants and \(n\) represents the degree of the polynomial.

Polynomial equations can have multiple variables and degrees, making them useful for representing complex relationships. Nonlinear systems of equations often include polynomial equations, requiring special attention and techniques to solve. These nonlinear systems cannot be graphed as a straight line.

• The degree of the equation tells us about the number of solutions it can have.
• To solve polynomial equations, techniques such as factoring, graphing, or numerical methods can be used.

Understanding the form and components of polynomial equations is crucial in solving them, as it allows for more straightforward manipulation and factorization.

Factoring Polynomials

Factoring polynomials is a critical technique used to simplify polynomial equations and solve them. The goal is to express a polynomial as a product of simpler polynomials or factors. This is especially helpful in setting equations to zero and then solving for the variable.

One common method is factoring by grouping, where terms are rearranged and grouped for factoring. Another method is the difference of squares, as seen in quadratic expressions like \(x^2 - 4\), which can be factored into \((x - 2)(x + 2)\).

• Start by identifying and pulling out the greatest common factor (GCF) if applicable.
• Next, look for patterns like the difference of squares, perfect square trinomials, or sum/difference of cubes.
• Factoring makes it easier to find solutions for the variable because it breaks the equation into manageable pieces.

In polynomial systems, factorization simplifies equations, allowing us to easily solve for roots. It is a toolset that connects closely with the substitution method, as it often prepares equations to interchange values effectively.

Substitution Method

The substitution method is used in solving systems of equations by replacing one variable with an expression obtained from another equation. This technique is especially handy in nonlinear systems, where it translates the given problem into a simpler form.

Here's how it works:

• Start by solving one of the equations for one variable in terms of the other variables. This step yields an expression that helps simplify the problem.
• Next, substitute this expression into the other equation(s). This step eliminates one variable, reducing the complexity of the system.
• Solve the resulting equation, typically a polynomial equation. Once the variable is found, back-substitute to find other variables.

Using the substitution method, we managed to turn a nonlinear system into a factored polynomial form, facilitating the discovery of variable values. The method breaks down the original problem into easier-to-handle pieces and directs the solution process step-by-step for clarity and simplicity.