Problem 36 - College Algebra with Modeling and Visualization

Question

Graph each system of equations and find any solutions. Check your answers. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. (GRAPH CAN'T COPY) $$ \begin{array}{c} x-4 y=4 \\ 2 x-8 y=4 \end{array} $$

Step-by-Step Solution

Answer

No solution; the system is inconsistent and independent.

Step 1: Rewrite Equations in Slope-Intercept Form

The first step is to convert each equation into the slope-intercept form \(y = mx + b\). For the equation \(x - 4y = 4\), rearrange the terms to isolate \(y\):\[x - 4y = 4 \implies y = \frac{1}{4}x - 1\]. For the second equation \(2x - 8y = 4\), isolate \(y\):\[2x - 8y = 4 \implies y = \frac{1}{4}x - \frac{1}{2}\].

Step 2: Graph the Equations

Using the slope-intercept form, plot each line on a graph. For \(y = \frac{1}{4}x - 1\), the y-intercept is -1, and the slope is \(\frac{1}{4}\). For \(y = \frac{1}{4}x - \frac{1}{2}\), the y-intercept is -\(\frac{1}{2}\), and the slope is \(\frac{1}{4}\). Both lines have the same slope but different intercepts, meaning they will be parallel lines.

Step 3: Determine Intersection and Consistency

Since the lines are parallel and have no points of intersection, the system has no solutions. A system with no points of intersection is inconsistent. Because the lines are parallel with distinct intercepts, they are not the same line and thus the equations are independent.

Step 4: Check the Solution

Visually on the graph, you can see the lines never intersect, confirming there is no solution. Algebraically, solving the system using substitution or elimination will show a contradiction, hence confirming inconsistency.

Key Concepts

Graphing Systems of Equations

Graphing systems of equations can be a straightforward method to find solutions, especially when dealing with two variables. By transforming each equation into the slope-intercept form, which is expressed as \(y = mx + b\), it becomes easier to plot them on a graph.
For a system like the one in our exercise, we start by rewriting the equations:

• From \(x - 4y = 4\), we derive \(y = \frac{1}{4}x - 1\).
• Similarly, from \(2x - 8y = 4\), we get \(y = \frac{1}{4}x - \frac{1}{2}\).

Once both equations are in this form, plot each line using their respective slopes and y-intercepts.
For our example, both lines share a slope of \(\frac{1}{4}\) but differ in their y-intercepts (-1 and -\(\frac{1}{2}\)). This will visually show whether the lines intersect. Parallel lines (as in this case) indicate no intersection, and thus, no solution.

Consistent and Inconsistent Systems

In the context of systems of equations, understanding whether a system is consistent or inconsistent is key. These terms refer to whether the lines represented by the equations intersect:

• Consistent systems have at least one solution because their lines intersect at one or more points.
• Inconsistent systems have no solutions as their lines are parallel and never meet.

In our example, both equations, when plotted, form parallel lines. Since they do not meet at any point, the system is inconsistent.
The graphical approach shows this clearly, but algebraic methods like substitution or elimination validate it, confirming that there is no possible solution that satisfies both equations simultaneously. Understanding these definitions helps in predicting solutions before full graphing or calculation is complete.

Dependent and Independent Equations

Equations can also be classified as dependent or independent. This classification relates to the relationship between the equations themselves:

• Dependent equations are those where one equation can be derived from the other; essentially, they represent the same line.
• Independent equations are distinct; no equation can be derived from the other using multiplication or division without remainder.

In our provided exercise, even though the lines have the same slope, they have different y-intercepts, showing that the equations are independent. This means that while the directional slope is identical, the starting points (intercepts) are different, creating parallel but not overlapping lines.
Identifying dependency is crucial for predicting the system's number of solutions and understanding its geometric representation.