Problem 59 - College Algebra with Modeling and Visualization

Question

If possible, solve the nonlinear system of equations. $$ \begin{aligned} x^{2}+y^{2} &=20 \\ y &=2 x \end{aligned} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the problem

We need to solve a system of nonlinear equations. The equations are: \( x^2 + y^2 = 20 \) and \( y = 2x \). These represent a circle and a line on the coordinate plane.

Step 2: Substitute the linear equation into the circle equation

Substitute \( y = 2x \) into the circle equation, \( x^2 + y^2 = 20 \), to eliminate \( y \). This gives \( x^2 + (2x)^2 = 20 \).

Step 3: Simplify the equation

Simplify the equation \( x^2 + (2x)^2 = 20 \) to get \( x^2 + 4x^2 = 20 \). This combines to \( 5x^2 = 20 \).

Step 4: Solve for \( x \)

Divide both sides of the equation \( 5x^2 = 20 \) by 5 to isolate \( x^2 \). This gives \( x^2 = 4 \). Taking the square root of both sides, we find \( x = 2 \) or \( x = -2 \).

Step 5: Solve for \( y \) using \( y = 2x \)

Using the solutions for \( x \), calculate \( y \) using \( y = 2x \). If \( x = 2 \), then \( y = 2(2) = 4 \). If \( x = -2 \), then \( y = 2(-2) = -4 \).

Step 6: Identify solutions

The solutions are the ordered pairs \((2, 4)\) and \((-2, -4)\). Each pair is a point where the line \( y = 2x \) intersects the circle \( x^2 + y^2 = 20 \).

Key Concepts

System of Equations

A system of equations involves solving for multiple variables by using more than one equation at the same time. In this case, we are dealing with two equations: \(x^2 + y^2 = 20\) and \(y = 2x\). These equations need to be solved together to find a common solution for the variables \(x\) and \(y\).
Combining these two equations helps us identify the points that satisfy both mathematical expressions. Here:

• The first equation, \(x^2 + y^2 = 20\), represents a circle with radius \(\sqrt{20}\).
• The second equation, \(y = 2x\), represents a straight line with a slope of 2.

Solving this system tells us at which points the line intersects the circle. In geometric terms, these are the points of intersection of the circle and the line on the graph.

Circle Equation

A circle equation is typically expressed in the form \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. In our current problem, \(x^2 + y^2 = 20\) represents a circle centered at the origin \((0,0)\) with a radius of \(\sqrt{20}\).
The standard circle equation illustrates the basic concept of a circle's geometry:

• The coefficients of \(x^2\) and \(y^2\) are both 1, indicating no stretching or skewing.
• The equation is set equal to the square of the circle's radius (\(\sqrt{20}\)), determining the scale of the circle.

Understanding circle equations is crucial when dealing with problems involving curves and radially symmetric objects. It allows you to visualize how algebraic expressions map onto geometrical figures.

Substitution Method

The substitution method is a technique used to solve systems of equations by replacing one variable with an expression derived from another equation. It's especially handy with nonlinear systems, where traditional elimination might be cumbersome.
Here's a step-by-step overview of how it was applied in our system:

• First, identify which equation can easily be rearranged. In our system, the linear equation \(y = 2x\) was chosen.
• Substitute this expression for \(y\) into the circle equation \(x^2 + y^2 = 20\). This changes it into a single-variable equation: \(x^2 + (2x)^2 = 20\).
• Simplify and solve the resulting single-variable equation, \(5x^2 = 20\), which can be solved to find \(x = 2\) or \(x = -2\).

This method helps consolidate the equations into one, mastering the art of isolating and solving for variables effectively, which is key in tackling complex mathematical problems.