Problem 54 - 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. $$\begin{array}{r} x-3 y=9 \\ 3 x-9 y=18 \end{array}$$

Step-by-Step Solution

Answer

The given linear equations form parallel lines when graphed, due to having the same slope \(\frac{1}{3}\).

Step 1: Rewrite the equations in slope-intercept form

The general form of a linear equation is \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. To compare the slopes of the two given equations, they should first be rewritten in this form. The first equation on solving for \(y\) becomes \(y=\frac{1}{3}x - 3\), and the second equation becomes \(y=\frac{1}{3}x - 2\).

Step 2: Observe and compare the slopes

From the slope-intercept form of the equations, it can be seen that the coefficients of \(x\) in both equations (which represent the slopes of the lines) are the same, namely, \(\frac{1}{3}\). This means that the lines are parallel, as parallel lines have equal slopes. The y-intercepts are different, thus the lines are not coincident (i.e., they do not overlap completely).

Step 3: Graph the lines

To confirm the analysis, plot the two lines on a rectangular coordinate system. When plotted, the two lines appear parallel, confirming mathematical conclusion.

Key Concepts

Parallel Lines

In geometry, two lines are considered parallel if they run alongside each other without ever meeting, no matter how far they are extended. This can be understood better in the context of linear equations, where parallel lines will have identical slopes. The slope is a measure of how steep a line is. For example, in our given linear equations, each line has a slope of \(\frac{1}{3}\).

• If two linear equations have the same slope, they are parallel.
• Parallel lines never intersect because they rise and run in perfect sync.

When analyzing equations, always consider their slopes to determine if the lines are parallel. Even if two lines have the same slope, their different y-intercepts mean they will never cross paths.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \(y = mx + b\). Here, \(m\) represents the slope and \(b\) indicates where the line crosses or intercepts the y-axis. This form is particularly useful because it immediately tells you two crucial bits of information about the line:

• The slope \(m\), which indicates direction and steepness.
• The y-intercept \(b\), which marks the starting point on the y-axis.

For practical purposes, converting an equation to slope-intercept form simplifies graphing and comparing different lines. For instance, by rearranging \(x - 3y = 9\) to \(y = \frac{1}{3}x - 3\), it's evident that both the slope and y-intercept can be quickly identified. When you practice converting equations, it enhances your ability to visualize and interpret linear graphs effectively.

Rectangular Coordinate System

A rectangular coordinate system is a two-dimensional plane used to plot points, lines, and curves. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis. Each point or object in this system is described using an ordered pair \((x, y)\).

• The x-coordinate specifies horizontal position from the y-axis.
• The y-coordinate specifies vertical position from the x-axis.

This system is essential for graphing linear equations because it visually represents mathematical relationships. In our example, once the equations are rewritten in slope-intercept form, they use this coordinate system to plot points. To do this, simply choose a range of x-values, substitute them into the linear equation to find corresponding y-values, and mark these points on your graph. Connecting these points will display your linear equation as a line.The rectangular coordinate system helps us see how equations behave in a visual space, providing clarity and aiding in understanding complex concepts.