Problem 19 - Algebra 2

Question

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(y=x+4\) \(y=x-4\)

Step-by-Step Solution

Answer

The system is inconsistent.

Step 1: Understand the System of Equations

The system consists of two linear equations. The first equation is presented as \( y = x + 4 \) and the second one as \( y = x - 4 \). Both lines have the same slope of 1 but different y-intercepts, 4 and -4 respectively.

Step 2: Graph the First Equation

Graph the first line \( y = x + 4 \). Start at the y-intercept (0, 4) and use the slope of 1 (which means rise over run is 1/1) to plot another point (1, 5). Draw a straight line through these points.

Step 3: Graph the Second Equation

Graph the second line \( y = x - 4 \). Begin at the y-intercept (0, -4) and use the slope of 1 to find another point (1, -3). Draw a line through these points.

Step 4: Analyze the Intersection of the Lines

Since both lines have the same slope but different y-intercepts, they are parallel and will never intersect.

Step 5: Classify the System

Because the lines are parallel and do not intersect, the system has no solution. Therefore, it is classified as inconsistent.

Key Concepts

Consistent vs Inconsistent Systems

Understanding whether a system of equations is consistent or inconsistent is key in solving them. A **consistent system** means that there is at least one solution where the equations intersect. This could be a single point (consistent and independent) or they could overlap entirely, meaning infinite solutions (consistent and dependent).
An **inconsistent system**, on the other hand, has no solutions. The equations do not intersect at any point. This typically happens when lines are parallel, as they never meet. In the exercise, the system is inconsistent because the two lines are parallel and never touch.
Parallel lines share the same slope and have different y-intercepts, reflecting why they don’t intersect. This good grasp of consistency is essential in identifying how equations relate in a graph.

Graphing Linear Equations

Graphing helps visualize relationships between equations in a system. A **linear equation** in the form of \( y = mx + b \) is simple to graph by following steps:

• Identify the slope \( m \) and the y-intercept \( b \).
• Start by plotting the y-intercept on the graph.
• Use the slope to determine the direction and steepness of the line. The slope is the rise over the run (\( \frac{\text{rise}}{\text{run}} \)).
• From the y-intercept, move up or down for the rise and left or right for the run to plot another point.
• Connect the dots to draw a straight line.

In the exercise, both lines are graphing according to these guidelines. The line \( y = x + 4 \) starts at the y-intercept (0, 4), while \( y = x - 4 \) begins at (0, -4). Consistency in graphing ensures accuracy in determining intersections or parallel nature.

Parallel Lines

Parallel lines are a crucial concept in understanding systems of linear equations. These lines have the same slope but different y-intercepts. This results in them never crossing each other, making them similar but distinct. This is visually represented when you graph the equations and notice they run alongside at all times.
Key characteristics of parallel lines include:

• Shared slope: For instance, in the equations \( y = x + 4 \) and \( y = x - 4 \), both have a slope of 1.
• Never intersecting: Different y-intercepts like 4 and -4 ensure the lines never meet.

Parallel lines are significant when determining if a system is consistent or inconsistent. As in our exercise, identifying parallel lines leads us to label it as an inconsistent system. Grasping this concept not only aids in classification but also in solving more complex systems.