Problem 55 - College Algebra with Modeling and Visualization

Question

If possible, solve the nonlinear system of equations. $$ \begin{aligned} x^{2}-y &=0 \\ 2 x+y &=0 \end{aligned} $$

Step-by-Step Solution

Answer

The solutions are (0, 0) and (-2, 4).

Step 1: Solve for y in the Linear Equation

We begin by solving the linear equation, \(2x + y = 0\). Solve for \(y\) to make it easier to substitute into the other equation. This gives \(y = -2x\).

Step 2: Substitute into the Nonlinear Equation

Next, substitute \(y = -2x\) into the nonlinear equation \(x^2 - y = 0\). This results in \(x^2 - (-2x) = 0\), which simplifies to \(x^2 + 2x = 0\).

Step 3: Solve for x

Solve \(x^2 + 2x = 0\) by factoring. This gives \(x(x + 2) = 0\). Setting each factor equal to zero gives \(x = 0\) or \(x = -2\).

Step 4: Solve for y using x-values

Using \(x = 0\): Substitute into \(y = -2x\) to get \(y = -2(0) = 0\). The solution is \((0, 0)\).Using \(x = -2\): Substitute into \(y = -2x\) to get \(y = -2(-2) = 4\). The solution is \((-2, 4)\).

Key Concepts

Solving Systems of Equations

When faced with a system of equations, the goal is to find the values of the variables that satisfy all equations simultaneously. In this particular case, we are dealing with a nonlinear system because one or more equations include non-linear terms, such as an equation containing an exponent like the quadratic term \(x^2\). Therefore, finding solutions involves a bit extra care compared to linear systems. Solving systems of equations typically involves several methods:

• Graphical Method: Visualize where the equations intersect on a graph.
• Substitution: Isolate one variable in an equation and substitute it into another.
• Elimination: Add or subtract equations to eliminate a variable.

Here, we focus on the substitution method, as it often simplifies solving nonlinear systems.

Factoring Equations

Factoring is a strategy to break down equations into simpler components, making them easier to solve. In our example, we encountered the equation \(x^2 + 2x = 0\). Factoring is crucial here because it allows us to identify the potential solutions by breaking the equation down: - First, we take out a common factor of \(x\).- This gives us the factored form \(x(x + 2) = 0\).By setting each factor equal to zero, we solve for \(x\):

• \(x = 0\)
• \(x + 2 = 0\) which simplifies to \(x = -2\)

Factoring simplifies the equation and provides straightforward solutions for the variables, revealing critical values that satisfy the original equation.

Substitution Method

The substitution method is a straightforward technique used to simplify a system of equations by substituting one equation into another. In the given exercise, we started with the equation \(2x + y = 0\). Our first step was to manipulate this equation to express one variable in terms of the other: \(y = -2x\).After solving for \(y\), we substituted \(y = -2x\) into the nonlinear equation \(x^2 - y = 0\). The new equation becomes \(x^2 + 2x = 0\). By substitution, we've reduced a potentially complex system into a more manageable expression, enabling easier problem-solving with the factoring method. This method is particularly useful when one of the equations is already simple enough to rearrange easily.

Linear Equations

A linear equation is any equation that generates a straight line when plotted on a graph. They typically have variables raised to the first power and are expressed in the form \(ax + by = c\). In our exercise, the equation \(2x + y = 0\) is linear and can be visualized as a straight line. Some valuable traits of linear equations include:

• They have constant rates of change, represented graphically by a straight line.
• Solving linear equations is usually more straightforward as there is no involvement of complex terms like exponents or roots.
• Linear equations can be easily arranged for substitution or elimination methods.

Recognizing the linear nature of an equation helps streamline the approach taken to solve a system, simplifying the problem into more manageable steps.