Problem 4 - 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) $$ \left(\begin{array}{c} 2 x-y=9 \\ 4 x-2 y=11 \end{array}\right) $$

Step-by-Step Solution

Answer

The system is inconsistent; no solution exists.

Step 1: Understand the System

We are given a system of two equations: \(2x - y = 9\) and \(4x - 2y = 11\). Our task is to use graphing to determine if the system is consistent, inconsistent, or dependent and find the solution if it is consistent.

Step 2: Rewrite Equations in Slope-Intercept Form

Convert each equation to the slope-intercept form \(y = mx + b\) to make graphing easier.1. For \(2x - y = 9\), solve for \(y\): \[ y = 2x - 9 \]2. For \(4x - 2y = 11\), solve for \(y\): \[ 2y = 4x - 11 \] \[ y = 2x - \frac{11}{2} \]

Step 3: Graph the Equations

Plot both equations on the graph using their slope-intercept forms:- The first equation \(y = 2x - 9\) has a y-intercept of -9 and a slope of 2.- The second equation \(y = 2x - \frac{11}{2}\) has a y-intercept of \(-\frac{11}{2}\) and the same slope of 2.Draw both lines on the graph.

Step 4: Analyze the Graph

Observe the graph: - Both lines have the same slope but different y-intercepts, which means they are parallel. - Since the lines do not intersect, the system is inconsistent.

Step 5: Conclude the System Type

Because the lines are parallel and do not meet at any point, the system of equations is inconsistent. Thus, there is no solution to this system.

Key Concepts

Graphing Method

The graphing method is a visual way to solve system of equations. It involves plotting each equation on the same graph to find out where they intersect. This intersection point, if it exists, is the solution to the system of equations.

Here’s how you can apply the graphing method:

• First, convert each equation into a format that is easy to graph, such as the slope-intercept form.
• Next, plot each equation on the graph using their slope and y-intercept.
• Finally, analyze where, if anywhere, the lines on the graph intersect:
• If the lines intersect at a point, the system is consistent and has one solution at that point.
• If the lines are parallel and never intersect, the system is inconsistent and has no solution.
• If the lines are coincident, they overlap entirely, meaning the system is dependent and has infinitely many solutions.

When using the graphing method, understanding how to graph lines accurately is crucial, as small errors can lead to incorrect conclusions about the system's solutions.

Inconsistent Systems

An inconsistent system of equations is one where there is no solution. This typically occurs when the equations represent parallel lines. Parallel lines never intersect, meaning there is no common solution for the system.

In the context of a graph:

• Inconsistent systems usually arise from lines that have the same slope but different y-intercepts.
• These characteristics mean the lines will never meet, as they run parallel to each other.

In practical terms, if you were solving a problem and you graph two lines that represent each equation and see that they never intersect, you have an inconsistent system. Recognizing an inconsistent system is important because it tells you that searching for a common solution is futile.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to express the equation so it is easy to graph. The form is written as \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, showing where the line crosses the y-axis.

The slope-intercept form simplifies the process of graphing because it directly provides the slope and the y-intercept. This means you can quickly plot the starting point on the y-axis (\( b \)) and use the slope (\( m \)) to find another point on the line.

When dealing with systems of equations, converting each equation to slope-intercept form makes it easier to visualize and compare their graphical relationships, such as parallel, intersecting, or coincident lines.