Problem 63 - College Algebra with Modeling and Visualization

Question

If possible, solve the nonlinear system of equations. $$ \begin{array}{rr} 2 x^{2}-y= & 5 \\ -4 x^{2}+2 y= & -10 \end{array} $$

Step-by-Step Solution

Answer

The system has infinitely many solutions: \( (x, y) = (x, 2x^2 - 5) \) for any real number \( x \).

Step 1: Analyze the Equations

We have a nonlinear system of equations:\[ 2x^2 - y = 5 \] and \[ -4x^2 + 2y = -10 \].First, let's check how to simplify or manipulate these equations to find a potential solution.

Step 2: Solve the First Equation for y

From the first equation \( 2x^2 - y = 5 \), solve for \( y \):\[ y = 2x^2 - 5. \]

Step 3: Substitute y into the Second Equation

Substitute \( y = 2x^2 - 5 \) into the second equation \( -4x^2 + 2y = -10 \):\[ -4x^2 + 2(2x^2 - 5) = -10. \]

Step 4: Simplify the Equation

Simplify the equation:\[-4x^2 + 4x^2 - 10 = -10.\]This simplifies to:\[-10 = -10.\]This indicates that the equation is valid for any \( x \).

Step 5: Analyze the Result

Since any \( x \) results in a valid equation, we have infinitely many solutions for \( x \). For each \( x \), \( y \) can be calculated from \( y = 2x^2 - 5 \).

Step 6: General Solution for the System

The system of equations has infinitely many solutions given by the pair \( (x, y) = (x, 2x^2 - 5) \) for any real number \( x \).

Key Concepts

Solution Methods

When tackling a nonlinear system of equations, there are several solution methods to consider. The primary approach is to use algebraic manipulation techniques to either eliminate one of the variables or rewrite the system in a more manageable form.
For nonlinear equations, such as equations involving squares or higher powers, there are a few common methods:

• Substitution: This involves solving one of the equations for one variable, then substituting this expression into the other equation. It reduces the system to a single equation with one variable, making it easier to solve.
• Graphical Method: By plotting both equations on a graph, the intersection points of the curves represent the solutions to the system. This method is visual but may not be precise without technology.
• Elimination: Similar to substitution, this method involves adding or subtracting the equations to eliminate one of the variables. It works well if the coefficients of one variable are easy to equalize by multiplication or division.

The best solution method often depends on the structure of the equations and the information needed. For systems resulting in simpler expressions like the one in the exercise, substitution is usually effective.

Infinitely Many Solutions

The concept of infinitely many solutions arises in a system of equations when the equations are not independent. In such cases, the system is said to be dependent, meaning each equation describes the same line or curve in two different forms.
When simplifying the basic form of these systems, they often resolve into a tautology, like \-10 = -10\, as in this exercise. This indicates that the system is valid for all values of the variable involved. For every possible input, there is a corresponding output that satisfies both equations.
In the current problem, after substitution and simplification, we determined that the original two equations effectively describe the same relation between \(x\) and \(y\). Therefore, the solution set is infinite, comprising all pairs \((x, 2x^2 - 5)\) where \(x\) is any real number.
Such outcomes are crucial in understanding that not all systems have unique or finite solutions. Recognizing these patterns helps in categorizing and solving more complex systems.

Substitution Method

The substitution method is an effective technique for solving systems of equations, especially when dealing with nonlinear equations. It involves a straightforward process where one begins by isolating one of the variables in one of the equations.
In this exercise, the first step involved solving the first equation for \(y\), giving:

• \(y = 2x^2 - 5\)

This expression is then substituted into the second equation, replacing \(y\) with \(2x^2 - 5\). This yields a single-variable equation that is easier to handle.
After substitution, simplifying the result allows you to understand the relation of one variable to another. In our problem, simplifying produced a tautology, indicating infinite solutions. The power of substitution lies in transforming complex systems into forms that reveal underlying truths about the relationships between variables.
While simple in principle, mastery of the substitution method requires practice, especially in identifying when it is the best option versus other possible strategies. Importantly, ensure the substitution and simplification steps are performed correctly to avoid errors that could lead to incorrect conclusions.