Problem 5 - Intermediate Algebra

Question

For Problems \(1-16\), use the graphing approach to determine whether the system is consistent, the system is inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. (Objective 1) $$ \left(\begin{array}{l} \frac{1}{2} x+\frac{1}{4} y=9 \\ 4 x+2 y=72 \end{array}\right) $$

Step-by-Step Solution

Answer

The system is dependent with infinitely many solutions.

Step 1: Convert Equations to Slope-Intercept Form

First, we need to transform each equation from the system into slope-intercept form (\(y = mx + b\)). For the first equation \(\frac{1}{2}x + \frac{1}{4}y = 9\), solve for \(y\):\[\frac{1}{4}y = -\frac{1}{2}x + 9\]\[y = -2x + 36\]. For the second equation \(4x + 2y = 72\), solve for \(y\): \[2y = -4x + 72\]\[y = -2x + 36\].

Step 2: Plot the Equations on a Graph

Now plot the two linear equations on a graph using their slope-intercept forms. Both equations are the same: \(y = -2x + 36\). Therefore, both lines will overlap.

Step 3: Analyze the Graph

Since both equations resulted in the same line, their graphs overlap completely. This indicates the system of equations is dependent, meaning they are essentially the same line, and there are infinitely many solutions.

Key Concepts

Graphical Solution of Equations

Graphing is a visual method to find the solution of a system of linear equations. It involves plotting the equations on a coordinate grid. Each equation will appear as a straight line thanks to their linear nature. The point where the lines intersect represents the solution of the system.

In our example, both equations are plotted on a graph. When transformed to slope-intercept form, they result in the same equation:

• \(y = -2x + 36\)

This overlapping means every point one line passes through, the other does as well. Such scenarios provide a graphical view of "dependent equations," where both lines are actually the same. Essentially, they represent the same mathematical relationship and visually overlap on the graph.

Graphical solutions make it easier to determine the nature of the system (whether consistent, inconsistent, or dependent) by simply observing how the lines interact. If two lines intersect at a single point, the solution is that point's coordinates. If they are parallel, no solution exists. And if they are the same line, there are infinite solutions.

Dependent Equations

Dependent equations describe a situation where two equations result in the same line when graphed. This means all points on one line are points on the other line as well.

The equations in dependent systems are essentially multiples of each other and describe the same geometric line in a coordinate plane. For example, transforming both equations,

• \(\frac{1}{2}x + \frac{1}{4}y = 9\)
• \(4x + 2y = 72\)

results in the identical line \(y = -2x + 36\). This confirms they are mathematically the same.

A dependent system implies an infinite number of solutions because any point on the line satisfies both equations. This illustrates that dependent equations in a system do not provide unique solutions but instead demonstrate a relationship that holds true over an infinite set of points.

Consistent and Inconsistent Systems

When analyzing systems of equations, understanding consistency is key. A consistent system has at least one solution, while an inconsistent system has none.

In a consistent system:

• If the lines intersect at one point, it has a unique solution.
• If the lines overlap entirely, it's a dependent system, resulting in infinitely many solutions.

In our exercise, since the lines coincide completely, it's consistent and dependent.

Inconsistent systems, by contrast, are characterized by parallel lines that never meet. They graphically manifest as two lines that have the same slope but different y-intercepts. This condition reflects no common points or solutions that satisfy both equations, thus making the system inconsistent.

Understanding these concepts is crucial for solving systems of equations effectively, whether graphically, algebraically, or arithmetically. By recognizing the relationships between the equations, you can predict the number of solutions or identify if solving is even required.