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

Step-by-Step Solution

Answer

The system is inconsistent; the lines are parallel and do not intersect.

Step 1: Understand the Equations

We are given two linear equations: \(2x - y = 9\) and \(4x - 2y = 11\). Our task is to determine whether the system is consistent, inconsistent, or dependent by graphing.

Step 2: Convert to Slope-Intercept Form

To graph the equations, it is helpful to convert them to the slope-intercept form \(y = mx + b\). For the first equation \(2x - y = 9\), rearrange to get \(y = 2x - 9\). For the second equation \(4x - 2y = 11\), rearrange to get \(2y = 4x - 11\) and divide by 2 to obtain \(y = 2x - \frac{11}{2}\).

Step 3: Draw the Graph

Plot both lines on the graph. The line \(y = 2x - 9\) has a slope of 2 and a y-intercept of -9. The line \(y = 2x - \frac{11}{2}\) also has a slope of 2 but a y-intercept of \(-\frac{11}{2}\). Since both lines have the same slope but different y-intercepts, they are parallel.

Step 4: Analyze the Graph

Because the lines are parallel, they never intersect. This means there is no solution where both equations are satisfied simultaneously.

Step 5: Determine the Consistency of the System

Since the lines are parallel and do not intersect, the system of equations is inconsistent. There is no point where the two lines meet.

Key Concepts

Linear Equations

Linear equations are like the building blocks of algebra. They are called "linear" because when you graph them, you get straight lines. A typical linear equation can look like this: \( ax + by = c \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables that change. These equations have two key parts: the slope and the intercept.

• **Slope**: This tells you how steep the line is. If it’s positive, the line goes up as you move to the right. If it’s negative, the line goes down.
• **Intercept**: This is where the line crosses the y-axis. It tells you what value \( y \) has when \( x \) is zero.

Understanding these components helps you graph the equation or see what it looks like on paper. In our example, when the equations are in slope-intercept form, you can easily identify the slope and intercept to make graphing straightforward.

Consistent and Inconsistent Systems

When talking about systems of equations, the term "consistent" means something very specific. It indicates whether the equations in the system share any common solutions, or points in common on a graph. Imagine two lines on a graph; their relationship determines the system's consistency.

• **Consistent Systems**: These have at least one solution. The lines on the graph will intersect at one point (one solution) or lie on top of each other (infinite solutions, meaning the equations are dependent).
• **Inconsistent Systems**: These have no solutions because the lines are parallel and never intersect, as they have the same slope but different y-intercepts.

If the lines were to cross, that crossing point would be the solution to both equations. But in our example, since the lines are parallel with no points of intersection, the system is inconsistent.

Slope-Intercept Form

Slope-intercept form is a way to write linear equations so they’re easy to understand and graph. The format is:\( y = mx + b \)
- \( m \) stands for the **slope** of the line. This value shows how much \( y \) increases when \( x \) increases by 1.
- \( b \) is the **y-intercept**. This is where the line crosses the y-axis.
Using slope-intercept form makes it straightforward to draw a line on a graph. By looking at \( m \), you know if the line will rise or fall as you go from left to right.
For the example equations given, we converted them into this form to better understand their properties. The first equation became \( y = 2x - 9 \), and the second was \( y = 2x - \frac{11}{2} \). Both lines have the same slope, 2, implying they're parallel and revealing that the system is inconsistent as they don’t share any solutions.