Problem 8 - Algebra for College Students

Question

Use the graphing approach to determine whether the system is consistent, the system in inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. \(\left(\begin{array}{l}4 x-9 y=-60 \\ \frac{1}{3} x-\frac{3}{4} y=-5\end{array}\right)\)

Step-by-Step Solution

Answer

The system is consistent and dependent; all points on the line are solutions.

Step 1: Rewrite Equations in Slope-Intercept Form

We start by converting each equation into the slope-intercept form, which is of the form \(y = mx + b\). This will make graphing the equations easier.For the first equation, \(4x - 9y = -60\):\[9y = 4x + 60 \quad \rightarrow \quad y = \frac{4}{9}x + \frac{60}{9} \quad \rightarrow \quad y = \frac{4}{9}x + \frac{20}{3}\]For the second equation, \(\frac{1}{3}x - \frac{3}{4}y = -5\):\[-\frac{3}{4}y = -\frac{1}{3}x - 5 \quad \rightarrow \quad y = \frac{4}{9}x + \frac{20}{3}\]

Step 2: Plot the Equations on a Graph

On the same set of axes, we plot both equations. Since both equations simplify to the same equation, \(y = \frac{4}{9}x + \frac{20}{3}\), they represent the same line on the graph. This indicates that they have infinitely many points in common.

Step 3: Determine the Nature of the System

Since the two equations are dependent (they represent the same line), the system is consistent and dependent. This means that there are infinitely many solutions that satisfy both equations.

Step 4: State the Solution

Because the system represents the same line, the solution set is not just a single point, but rather every point on the line \(y = \frac{4}{9}x + \frac{20}{3}\).

Key Concepts

Consistent Systems

When dealing with systems of equations, we often need to determine whether a system is consistent or inconsistent. A consistent system is one where there is at least one solution. This means that the lines represented by the equations intersect at one or multiple points. In our example, the system is consistent because both equations graph to the same line.

Here’s a quick guide to identifying consistent systems:

• **Consistent and Independent**: The equations have a unique point of intersection, meaning the lines cross at exactly one point.
• **Consistent and Dependent**: The equations represent the same line, resulting in infinitely many intersections because they overlap completely.

In a consistent system such as our example, when graphing the equations, you'll notice they don't diverge away from each other — they show complete overlap or intersect at least once.

Dependent Equations

Dependent equations are a part of some consistent systems where both equations essentially describe the same line. This occurs when two or more equations in a system give equivalent expressions for the same relationship. In the given system, after converting both equations to the slope-intercept form, they become identical.

Understanding dependent equations involves recognizing a few key features:

• **Same slope and y-intercept**: If two equations in a system have both the same slope and y-intercept after simplification, they are dependent.
• **Infinite solutions**: Dependent systems have infinitely many solutions since each point on the shared line satisfies both equations.

In mathematical problems, identifying dependent equations can help in understanding the nature of the system, as it reveals that the concepts represented by the equations are not distinct but instead provide the same information.

Slope-Intercept Form

The slope-intercept form of an equation is a way of expressing linear equations. It’s written as \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept, the point where the line crosses the y-axis.
Converting equations to the slope-intercept form simplifies the process of graphing and understanding linear relationships.

• **Slope \(m\)**: The slope determines the steepness and direction of the line. A positive slope rises from left to right, while a negative slope falls.
• **Y-intercept \(b\)**: This is the value of \(y\) when \(x = 0\). It's where the line touches the y-axis.

In our original exercise, converting each equation to slope-intercept form made it clear that both lines have exactly the same slope and intercept, thus revealing their dependent nature. This form is especially handy because it gives a direct way to identify and graph a line.