Problem 32 - Precalculus with Limits

Question

Solving a System of Equations Graphically in Exercises \(33-44,\) solve the system graphically. $$\left\\{\begin{array}{l}{y=-x} \\ {y=x^{3}+3 x^{2}+2 x}\end{array}\right.$$

Step-by-Step Solution

Answer

The solutions to the system of equations are the points where the graphs of the two equations intersect. These exact points will be determined by finding the intersection points of the graphs.

Step 1: Graphing the linear equation

First graph the linear equation \(y=-x\). Since the slope of this line is -1 and the y-intercept is 0 (as there is no constant term), it is a straight line crossing the origin and falling towards the right. For every positive value of x, y is negative and vice versa.

Step 2: Graphing the cubic equation

Next, graph the cubic equation \(y=x^{3}+3x^{2}+2x\). This is a more complex function, curving based on the power of x. For negative values of x, y will be negative. As x increases, the value of y also increases, but at an accelerating rate.

Step 3: Determine the intersection points

Now, look for the points where the two graphs intersect. These intersection points represent the values of x and y that satisfy both equations simultaneously and are therefore the solutions to the given system of equations.

Key Concepts

Graphical Method

The graphical method is an intuitive way to solve systems of equations. It involves sketching the graphs of the equations on a coordinate plane and identifying their intersection points. These points represent the solutions to the system, where both equations hold true simultaneously.

To start, graph each equation separately. Ensure you have labeled axes and a scale that accurately represents both equations. Visualize the behavior of each graph to predict how and where they might meet.

This method provides a visual representation of solutions, helping to understand the relationships between equations and their roots. It is especially useful for understanding more complex equations like polynomials.

Linear Equation

A linear equation is one of the simplest forms of equations and is represented as a straight line on a graph. The general form is \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.

In our exercise, the linear equation is \( y = -x \), meaning it has a slope of -1 and a y-intercept of 0. This causes the line to pass through the origin and slant downwards towards the right as the slope is negative.

Key characteristics of linear equations include:

• They form a straight line.
• The slope determines the angle and direction of the line.
• It can be easily sketched with just two points.

By plotting this on a graph, you can clearly see how it interacts with other equations.

Cubic Equation

A cubic equation is more complex than a linear equation, characterized by the highest power of the variable being three. In general form, it is written as \( y = ax^3 + bx^2 + cx + d \).

In our given problem, the cubic equation is \( y = x^3 + 3x^2 + 2x \). It showcases increasing complexity with three terms involving powers of x. The nature of cubes and squares in the equation affects its graph into curves.

When graphing, expect some or all of the following:

• Curved lines that may change direction.
• Potential inflection points where the graph changes concavity.
• It might cross the x-axis or y-axis multiple times.

This equation curves upward as x values increase, showing accelerated growth due to the cubic term.

Intersection Points

Intersection points are crucial for solving systems graphically, as they indicate solutions fulfilling all set equations. These are the points where two or more graphs meet on the coordinate plane.

For linear and cubic graphs, these points can be found by visually observing the graph or by finding precise values using algebraic methods such as substitution or elimination.

Identifying intersection points helps in:

• Understanding the common solutions between different equations.
• Visualizing how different types of equations interact.
• Confirming that solutions satisfy all equations in the system.

In the example exercise, the intersection points give you the exact x and y values satisfying both the linear and cubic equation simultaneously.