Problem 42 - College Algebra

Question

Graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions. $$ \begin{array}{r} -x+2 y=4 \\ 2 x-4 y=1 \end{array} $$

Step-by-Step Solution

Answer

The system is inconsistent with no solution.

Step 1: Convert Equations to Slope-Intercept Form

The first step is converting both equations in the system to the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Start with the equation \(-x + 2y = 4\).\\[-x + 2y = 4 \Rightarrow 2y = x + 4 \Rightarrow y = \frac{1}{2}x + 2\].\Next, convert the equation \(2x - 4y = 1\) to slope-intercept form.\\[2x - 4y = 1 \Rightarrow -4y = -2x + 1 \Rightarrow y = \frac{1}{2}x - \frac{1}{4}\].

Step 2: Graph the Equations

Draw a coordinate plane and plot the graph for each equation. The line of the first equation \(y = \frac{1}{2}x + 2\) has a y-intercept of 2 and a slope of \(\frac{1}{2}\), which means for every 2 units move in the x-direction, the line moves up 1 unit. Similarly, the line \(y = \frac{1}{2}x - \frac{1}{4}\) has a y-intercept of \(-\frac{1}{4}\) and also a slope of \(\frac{1}{2}\). Plot at least two points for each and draw the lines.

Step 3: Analyze the Graph and Solution

Observe the lines you have drawn. Both lines have the same slope (\(\frac{1}{2}\)) but different y-intercepts (2 and -\(\frac{1}{4}\)), which means they are parallel and never intersect. Since the lines do not intersect, there is no solution to the system.

Step 4: Determine the System's Classification

Based on the graph analysis, the lines being parallel indicates that the system is inconsistent because there is no shared point between the lines. Therefore, it has no solution. Parallel lines imply that the equations do not represent the same line nor have a meeting point, confirming the system is neither consistent with one solution nor dependent.

Key Concepts

Slope-Intercept Form

When working with linear equations, one of the most helpful forms to use is the slope-intercept form. This form is written as \(y = mx + b\), where \(m\) represents the slope, and \(b\) represents the y-intercept of the line. By expressing an equation in this form, it becomes simpler to identify how the line behaves and where it crosses the y-axis.

To convert an equation into the slope-intercept form, you follow these steps:

• Solve for \(y\) on one side of the equation.
• Rearrange the equation so that it becomes \(y = mx + b\).

For example, with the equation \(-x + 2y = 4\), rearranging it gives you \(y = \frac{1}{2}x + 2\). This shows you that the slope \((m)\) is \(\frac{1}{2}\), indicating the line rises 1 unit for every 2 units it goes to the right. The y-intercept \((b)\) is 2, meaning the line crosses the y-axis at \(y = 2\).

Using the slope-intercept form not only makes it easier to graph lines but also aids in comparing them to determine their relationship.

Graphing Linear Equations

Once you have your equations in slope-intercept form, it's time to graph them. This is a visual way to understand how the equations relate to each other.

To graph a linear equation, particularly in the slope-intercept form \(y = mx + b\), follow these steps:

• Identify the y-intercept \(b\) and plot it on the y-axis.
• Use the slope \(m\), represented as a fraction \(\frac{rise}{run}\), to determine another point.
• Draw the line that connects these points, extending it across the graph.

For example, given \(y = \frac{1}{2}x + 2\), start by plotting the point (0, 2) on the y-axis. From there, go up 1 unit and 2 units to the right to find another point at (2, 3). Draw a straight line through these points.

Repeat this process for the other equation, \(y = \frac{1}{2}x - \frac{1}{4}\). This time, although the slope is the same, the y-intercept is different at \(-\frac{1}{4}\). This means, despite having the same slope, the lines are parallel and will not cross each other on the graph.

Consistent and Inconsistent Systems

When analyzing systems of equations, it's essential to determine whether they are consistent or inconsistent. This tells us how many solutions the system has.

- A **consistent system** has at least one solution. This occurs when the lines intersect, either at one point (indicating one solution), or completely overlap (indicating infinite solutions, making it dependent).- An **inconsistent system** has no solution. This happens when the lines are parallel, as they will never meet.

In the example provided, both lines have the same slope \((\frac{1}{2})\) but different y-intercepts (2 and \(-\frac{1}{4}\)). This makes them parallel, never touching, hence the system is inconsistent. Understanding these classifications helps in predicting the outcome of systems of equations without needing exhaustive computing of rearrangements or solving.