Problem 52 - Algebra 2

Question

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(y=x+2\) \(y=x-1\)

Step-by-Step Solution

Answer

Inconsistent; the system has no solutions as the lines are parallel.

Step 1: Identify the Equations

The system is given by two equations: \( y = x + 2 \) and \( y = x - 1 \). Both are linear equations in the slope-intercept form \( y = mx + b \).

Step 2: Analyze the Slopes

The slope-intercept form \( y = mx + b \) shows that the slope \( m \) is the same in both equations \( m = 1 \). Equivalent slopes mean the lines are parallel or the same line.

Step 3: Analyze the Intercepts

The y-intercepts of the lines are different. For \( y = x + 2 \), the y-intercept is 2. For \( y = x - 1 \), the y-intercept is -1. This indicates the lines are not the same.

Step 4: Graph the Equations

Graph both lines to confirm they are parallel. The first line passes through points like (0,2) and (1,3), and the second line passes through points like (0,-1) and (1,0).

Step 5: Classification of the System

Since the lines are parallel and do not meet, the system is classified as inconsistent. Inconsistent systems have no solutions.

Key Concepts

Linear Equations

Linear equations are a type of mathematical equation where the highest power of the variable is one. They create straight lines when graphed on a coordinate plane. Essential components of a linear equation include the slope and the y-intercept:

• The slope tells us how steep the line is, describing the change in the y-coordinate for a one-unit increase in the x-coordinate.
• The y-intercept is the point where the line crosses the y-axis.

Linear equations often appear in the form of slope-intercept form, written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. By identifying these components, we can easily graph and understand the behavior of linear equations.

Parallel Lines

Parallel lines are lines in a plane that never intersect. Even if extended infinitely, they stay the same distance apart and never cross each other. In the context of linear equations, lines are parallel if they have the same slope:

• If two lines have identical slopes but different y-intercepts, they are parallel.
• This characteristic implies that, despite having the same angle of inclination, the lines are offset vertically, which means they run alongside each other without ever meeting.

For example, the lines described by the equations \(y = x + 2\) and \(y = x - 1\) are parallel because both have a slope of 1. But their y-intercepts, 2 and -1, respectively, ensure they never intersect on the graph.

Inconsistent Systems

An inconsistent system of equations happens when there are no solutions that satisfy all equations in a given system. In the case of linear equations, this usually occurs when the lines are parallel:

• Parallel lines represent different equations with the same slope but different y-intercepts.
• Because they never intersect, there's no single point (x, y) that can satisfy both equations simultaneously.

In our original exercise with the equations \(y = x + 2\) and \(y = x - 1\), where the lines never meet, this is a classic example of an inconsistent system. When graphing such systems, you will see lines running in parallel which clearly indicates there is no common solution.

Slope-Intercept Form

The slope-intercept form of a linear equation is an easy way to graph a line and understand its characteristics. It is written as \(y = mx + b\), where:

• \(m\) is the slope, indicating the steepness or incline of the line. This tells you how much the y-value changes for every unit change in the x-value.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis. It tells you the value of y when x is 0.

This form is particularly useful because it directly shows both the slope and the y-intercept, making it straightforward to graph and understand a line's growth or decline. In our example, identifying the equations \(y = x + 2\) and \(y = x - 1\) as being in slope-intercept form immediately shows they are parallel, as they share the same slope, \(m = 1\).