Problem 52 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(47-56,\) graph both linear equations in the same rectangular coordinate system. If the lines are parallel or perpendicular, explain why. $$\begin{aligned} &y=x+2\\\ &y=-x-1 \end{aligned}$$

Step-by-Step Solution

Answer

The lines defined by the equations \(y=x+2\) and \(y=-x-1\) are neither parallel nor perpendicular when graphed on the same coordinate system.

Step 1: Understanding the Given Equations

The given equations are of the form \(y = mx + c\), where \(m\) is the slope of the line and \(c\) is the y-intercept. The two lines here are \(y = x + 2\) and \(y = -x - 1\). The slope of the first line is 1, and its y-intercept is 2. The slope of the second line is -1, and its y-intercept is -1.

Step 2: Graphing the Lines

Plot both lines on the same coordinate system. Start by plotting the y-intercepts (2 for the first line and -1 for the second line). Then, use the slopes to determine another point on each line (from the point (0,2) on the first line, increase y by 1 for every 1 increase in x. From the point (0,-1) on the second line, decrease y by 1 for every 1 increase in x). Connect these points to form the lines.

Step 3: Determine If Lines are Parallel or Perpendicular

Examine the relationship between the slopes of the two lines. Two lines are parallel if their slopes are the same, and they are perpendicular if the slopes are negative reciprocals of each other. Here, the slopes are 1 and -1, which are not equal and not negative reciprocals. Therefore, the lines are neither parallel nor perpendicular.

Key Concepts

Coordinate System

A coordinate system is a method for identifying the position of points on a plane using two numbers. This system is called a plane because it is flat, much like a piece of graph paper you might find in school. The most common coordinate system is the Cartesian coordinate system, which uses a horizontal axis, the x-axis, and a vertical axis, the y-axis.

When you place a point in this coordinate system, you'll give it two numbers (x, y). The first number tells you how far to move horizontally, while the second tells you how far to move vertically. This way, each point on the plane gets a unique address, much like a street address in real life. This concept is very helpful when graphing linear equations because it allows us to draw precise lines.

• The x-axis runs left and right.
• The y-axis runs up and down.
• Points are given as (x, y).

Understanding the coordinate system is key to graphing and analyzing linear equations.

Slopes

The slope of a line is a measure of its steepness. In mathematical terms, the slope is the ratio of the change in vertical distance (y) to the change in horizontal distance (x). Essentially, the slope tells you how much y changes as x changes, which is why it is often described as "rise over run."

For the line equation of the form \(y = mx + c\):

• \(m\) signifies the slope of the line.
• A higher value of \(m\) means a steeper slope.
• If \(m\) is positive, the line slopes upward as you move from left to right.
• If \(m\) is negative, the line slopes downward as you move from left to right.

The two lines given in the exercise have slopes of 1 and -1. These indicate that one line rises at a 45-degree angle (slope of 1), while the other falls at a 45-degree angle (slope of -1). The differences in slopes can tell us a great deal about how lines relate to each other geometrically.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. This is the value of y when x is zero, so it gives the starting point of the line on any graph.

In the equation \(y = mx + c\):

• \(c\) represents the y-intercept.
• For the line \(y = x + 2\), the y-intercept is 2, meaning the line crosses the y-axis at (0, 2).
• For the line \(y = -x - 1\), the y-intercept is -1, meaning it crosses the y-axis at (0, -1).

The y-intercept is useful because it provides a clear starting point when sketching graphs. By plotting the y-intercept first, you can use the slope to determine the direction and angle of the line from there.

Graphing

Graphing is the process of drawing a visual representation of mathematical equations. It serves as a useful way to understand the relationships between variables and visualize how equations behave in a coordinate system.

When graphing linear equations like \(y = x + 2\) and \(y = -x - 1\):

• Start by plotting the y-intercept on the graph.
• Use the slope to find another point: move up/down and left/right as necessary.
• Draw a line through your points to extend across the graph.

In this particular exercise, after plotting the points derived from the slopes and intersections, the resultant lines show their behavior and relationships. The visual clarity of the graph highlights how the lines don't intersect at right angles (nor are they parallel), supporting our mathematical analysis of their slopes.