Problem 8 - 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) $$ \left(\begin{array}{l} 4 x-9 y=-60 \\ \frac{1}{3} x-\frac{3}{4} y=-5 \end{array}\right) $$

Step-by-Step Solution

Answer

The system is dependent with infinitely many solutions.

Step 1: Convert Equations to Slope-Intercept Form

Start by solving each equation for y in terms of x. For the first equation, \(4x - 9y = -60\), solve for y: \[9y = 4x + 60 \\Rightarrow y = \frac{4}{9}x + \frac{60}{9} \\Rightarrow y = \frac{4}{9}x + \frac{20}{3} \]For the second equation, \(\frac{1}{3}x - \frac{3}{4}y = -5\), solve for y:\[-\frac{3}{4}y = -\frac{1}{3}x - 5 \\Rightarrow y = \frac{4}{9}x + \frac{20}{3} \]This results in the same equation, indicating the lines are dependent.

Step 2: Graph the Equations

Plot both equations on a graph based on their slope-intercept form. Given both resolved into: \(y = \frac{4}{9}x + \frac{20}{3}\), we notice that they are identical. This means they plot as the same line on the graph.

Step 3: Determine the Type of System

Since the equations are the same (\(4x - 9y = -60\) and \(\frac{1}{3}x - \frac{3}{4}y = -5\) simplify to the same line), the system is dependent, indicating infinitely many solutions.

Step 4: Verify the System Type

To confirm dependency, check if both equations derive from equivalent manipulation. Simplification from Step 1 showed both lines graph identically, confirming the infinite solutions as all points on the line satisfy both equations.

Key Concepts

Graphing Method

The graphing method involves plotting the equations of a system on a graph to visually identify the relationship between the lines. This approach helps in determining whether a system is consistent (has solutions), inconsistent (has no solutions), or dependent (equations represent the same line). To use the graphing method, follow these steps:

• Convert each equation to the slope-intercept form, making it easier to identify the slope and y-intercept.
• Plot each equation on the same graph, using the slope and y-intercept.
• Analyze the graph to see how the lines relate to one another.

If the lines intersect at a point, the system is consistent and has a unique solution. If the lines are parallel and never meet, the system is inconsistent with no solutions. And if the lines coincide, meaning they lie on top of each other, the system is dependent with infinitely many solutions.

Dependent Equations

Dependent equations in a system occur when two linear equations represent the same line on a graph. This means that both equations are essentially different expressions of the same line, leading to infinitely many solutions. To identify dependent equations, follow these steps:

• Convert each equation to the slope-intercept form. This helps in directly comparing the slopes and y-intercepts.
• If the resulting equations have identical slopes and y-intercepts, they are dependent, meaning they describe the same line.

In such cases, any point on the line will satisfy both equations simultaneously. This implies that there's not just one single solution, but rather an entire line of solutions. Therefore, when solving systems of linear equations, recognizing dependent equations is key in understanding the nature of the solution set.

Slope-Intercept Form

The slope-intercept form of a linear equation is crucial for easily graphing a line and understanding its characteristics. The general formula for the slope-intercept form is:\[ y = mx + b \]Here, \(m\) represents the slope of the line, which indicates the steepness. Alternatively, \(b\) is the y-intercept, the point where the line crosses the y-axis. To convert an equation to slope-intercept form:

• Solve the equation for \(y\) to isolate it on one side. This often involves rearranging terms and simplifying the equation.
• Identify \(m\) and \(b\) from the equation. This will allow you to quickly plot the line on a graph.

This form is particularly useful when comparing equations to identify consistent, inconsistent, or dependent systems, because it makes the relationship between lines clear through their slopes and y-intercepts. Using slope-intercept form simplifies the graphing method and assists in visualizing the solutions of the system.