Problem 65 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

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

Step 1: Write down the given equations

We have the system of equations: 1. \( x^2 - y = 4 \) 2. \( x^2 + y = 4 \). Our goal is to find values of \( x \) and \( y \) that satisfy both equations simultaneously.

Step 2: Add the equations

To eliminate \( y \), add the two equations: \((x^2 - y) + (x^2 + y) = 4 + 4\). This simplifies to \( 2x^2 = 8 \).

Step 3: Solve for x

Divide both sides of the equation \( 2x^2 = 8 \) by 2: \( x^2 = 4 \). Take the square root of both sides to get: \( x = 2 \) or \( x = -2 \).

Step 4: Substitute x back into one of the original equations

Use the first equation \( x^2 - y = 4 \) to solve for \( y \). First, substitute \( x = 2 \) into \( x^2 - y = 4 \): \( 4 - y = 4 \). Solve for \( y \), which gives \( y = 0 \). Now, substitute \( x = -2 \): \((-2)^2 - y = 4\) simplifies to \(4 - y = 4\). This also gives \( y = 0 \).

Step 5: List possible solutions

The possible solutions for the system are \((x, y) = (2, 0)\) and \((x, y) = (-2, 0)\). Both satisfy the original equations.

Key Concepts

Solving Systems of Equations

When working with systems of equations, the objective is to find a set of values for the unknown variables that satisfy all the equations at once. In the case of nonlinear systems, like the one given in this exercise, at least one of the equations is not linear.
You often encounter equations such as quadratics, polynomials, or others that form curves rather than straight lines when graphed. Solving these systems typically requires more advanced techniques than those used for linear systems, as the solution involves curves that intersect at specific points.
There are several methods to solve systems of equations:

• Substitution: Solve for one variable in terms of the other in one equation, then substitute this expression into the other equation.
• Elimination: Alter the equations so that one variable is canceled out when the equations are combined.
• Graphically: Plot the equations on a graph to find the intersection points, although this is less precise for exact solutions.

Choosing the most suitable method depends on the specific equations and the desired solution method.

Elimination Method

The elimination method is often used when we want to remove one variable from the system. By adding or subtracting the equations, we can "eliminate" one of the variables. This is especially useful if comparable coefficients exist or can be easily manipulated through multiplication.
In our example, adding the equations \(x^2 - y = 4\) and \(x^2 + y = 4\) results in the elimination of the \(y\) terms.
What remains is a simpler equation in just one variable: \(2x^2 = 8\). By solving this simplified equation, we immediately find possible values for \(x\), which can then be substituted back to find corresponding \(y\) values.
This method is powerful for systems where direct substitution can be cumbersome or when coefficients conveniently align to cancel a variable.

Substitution Method

Substitution involves solving one of the equations for a variable and substituting that solution into the other equation(s). It simplifies the system by reducing the number of variables in the remaining equations.
For instance, in systems where one equation can be easily rearranged, such as \(x^2 + y = 4\), solving for \(y\) gives \(y = 4 - x^2\).
While our current example utilized elimination primarily, substitution plays a crucial role in re-evaluations. Once \(x\) values are known, substitute them back into either original equation and swiftly find \(y\). This dual method approach ensures all potential solutions are covered.

Quadratic Equations

A quadratic equation is any equation that can be expressed in the form \(ax^2 + bx + c = 0\), where \(a eq 0\). Quadratics are prevalent in nonlinear systems and are characterized by their parabolic graphs.
In our system, both equations hosted the quadratic term \(x^2\), underlining their identity as quadratics. Because quadratics have up to two real roots, they can produce two valid sets of solutions.
The standard approaches to solving quadratic equations, such as factorizing, completing the square, and using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), are integral in tackling systems that include quadratic elements.
Understanding their properties enables effective problem-solving and provides insights, such as expecting two solutions for \(x\) or identifying symmetrical properties due to the square term.