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Chapter 2: Problem 56

Determine whether the graphs of each pair of equations are parallel, perpendicular, or neither. $$ 2 x+3 y=9,3 x-2 y=5 $$

Short Answer

Expert verified
The graphs are perpendicular because their slopes are opposite reciprocals.

Step by step solution

01

Rewrite the first equation in slope-intercept form

The first equation is \(2x + 3y = 9\). To convert this to the slope-intercept form \(y = mx + b\), solve for \(y\). \[2x + 3y = 9\] Subtract \(2x\) from both sides: \[3y = -2x + 9\] Now, divide each term by 3 to solve for \(y\): \[y = -\frac{2}{3}x + 3\] In this equation, the slope \(m\) is \(-\frac{2}{3}\).
02

Rewrite the second equation in slope-intercept form

The second equation is \(3x - 2y = 5\). Solve for \(y\) to convert it to slope-intercept form. \[3x - 2y = 5\] Subtract \(3x\) from both sides: \[-2y = -3x + 5\] Now, divide each term by -2: \[y = \frac{3}{2}x - \frac{5}{2}\] In this equation, the slope \(m\) is \(\frac{3}{2}\).
03

Compare the slopes to determine the relationship between the lines

The slope of the first line is \(-\frac{2}{3}\), and the slope of the second line is \(\frac{3}{2}\). Two lines are parallel if their slopes are equal, and they are perpendicular if the product of their slopes is \(-1\).
04

Check for perpendicularity

Calculate the product of the slopes: \(m_1 \times m_2 = -\frac{2}{3} \times \frac{3}{2}\). The product is \(-1\), which means these slopes are opposite reciprocals. Thus, the graphs of these equations are perpendicular.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form of a linear equation is denoted as \(y = mx + b\), where \(m\) represents the slope, and \(b\) represents the y-intercept. The slope \(m\) determines the steepness of the line, showing how much \(y\) changes with a change in \(x\). Meanwhile, the y-intercept \(b\) is the point where the line crosses the y-axis. Knowing how to convert a linear equation into this form is crucial. It allows you to quickly determine and graph the line based on its slope and y-intercept.
  • First, isolate \(y\) on one side of the equation.
  • Ensure that \(y\) is by itself, then express the remaining terms in the form \(mx+b\).
This transformation is vital because it simplifies the process of assessing the relationship between two linear equations.
Parallel Lines
Parallel lines have the distinct characteristic of having the same slope \(m\). This means they never intersect, no matter how far they are extended on a graph. If you see two lines with equal slopes in their slope-intercept forms, you can be sure they are parallel.
  • You'll often encounter parallel lines in problems involving systems of equations.
  • Parallel lines demonstrate a consistent rate of change without intersecting.
Recognizing parallel lines is crucial in both geometry and algebra when interpreting problems and constructing graphs.
Perpendicular Lines
Perpendicular lines intersect at a right angle (90 degrees). A key feature of perpendicular lines is the relationship between their slopes; they are opposite reciprocals. If you multiply the slopes of two perpendicular lines, the product is \(-1\).
  • If the slope of one line is \(\frac{a}{b}\), the slope of another line that is perpendicular will be \(-\frac{b}{a}\).
  • This opposite reciprocal relationship is a quick check for perpendicularity.
In mathematical problems, once you determine two lines are perpendicular, it gives you information about angles and distances in a given space.
Graphing Linear Equations
Graphing linear equations involves plotting points that line up to form a straight line. Once in the slope-intercept form, \(y=mx+b\), plotting becomes simple:
  • Start at the y-intercept \(b\) on the y-axis.
  • From there, use the slope \(m\) to identify additional points. For a slope of \(\frac{3}{2}\), it means rising 3 units for every 2 units you move right.
Connect these points to form the line. This visual representation helps in understanding the linear equation's behavior and any relationships with other lines on a graph. Consistent practice in graphing ensures strong foundations in recognizing linear relationships.

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