Problem 5 - 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. \(\left(\begin{array}{l}\frac{1}{2} x+\frac{1}{4} y=9 \\ 4 x+2 y=72\end{array}\right)\)

Step-by-Step Solution

Answer

The system is consistent and dependent; there are infinitely many solutions.

Step 1: Graph the First Equation

To graph the equation \(\frac{1}{2}x + \frac{1}{4}y = 9\), we first express \(y\) in terms of \(x\). Multiply everything by 4 to eliminate the fractions: \(2x + y = 36\). Now solve for \(y: y = -2x + 36\). Identify the y-intercept \((0, 36)\) and the slope \(-2\) to draw the line.

Step 2: Graph the Second Equation

For the second equation \(4x + 2y = 72\), simplify it by dividing every term by 2 to get \(2x + y = 36\). This can be rewritten as \(y = -2x + 36\). Identify the y-intercept \((0, 36)\) and the slope \(-2\) to draw the line.

Step 3: Compare the Graphs

Upon graphing both equations, we observe that they represent the same line, \(y = -2x + 36\). The equations are dependent, meaning all solutions to the first equation are also solutions to the second equation.

Step 4: Check the System Consistency

Since the equations graph as the same line, the system is consistent and dependent. There are infinitely many solutions to this system, represented by the line \(y = -2x + 36\). Any point on this line is a solution to the system.

Key Concepts

Consistent System

A consistent system of equations is characterized by having at least one solution. Such systems can typically intersect at a point or overlap completely as a single line. This means that there is some commonality between the equations that allows for the solution(s).

In the given exercise, the system of equations is graphed and found to be consistent. This occurs because both equations simplify to the same line, showing their inherent consistency. When graphing systems of equations, consistent systems can be identified as either intersecting at a single point (for independent systems) or forming one line (for dependent systems).

• A consistent system always has solutions.
• The graph of consistent systems will either intersect or coincide entirely.

Understanding consistent systems is key to solving equations as it informs if and how many solutions exist.

Dependent Equations

Dependent equations occur when two equations describe the exact same line on a graph. This means that one equation is simply a multiple or a rearrangement of the other.

In our original exercise, both equations yield the same line, indicating that the equations are dependent. Simplifying each gives us the linear form: \[y = -2x + 36\]When graphed, dependent equations overlap completely, indicating an infinite number of solutions to the system.

• Dependent equations are identical or scalar multiples of each other.
• They graph as the same line.
• There are infinitely many solutions because every point on the line satisfies both equations.

Understanding dependent equations helps explain why some systems have infinite solutions.

Solution Set

The solution set of a system of equations is the collection of all points that satisfy all equations simultaneously. In a consistent and dependent system, as in our example, the solution set is all the points along the line resulting from the overlap of the equations.

For the equations given:\[ rac{1}{2}x + rac{1}{4}y = 9 \]
and\[ 4x + 2y = 72 \]
these simplify to the same equation \[ y = -2x + 36 \]This line itself represents the infinite solution set for the system. Every coordinate point along this line is a valid solution.

• A solution set contains all solutions for the equations.
• For dependent equations, it typically describes a whole line.
• Checking the solution set visually via graphing helps verify the consistency and dependency of the system.

Solution sets provide a numerical description of where and how systems of equations overlap, and thus, are crucial for comprehensive mathematical understanding.