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Chapter 15: Problem 54

Determine whether each pair of lines is parallel, perpendicular, or neither. \(y=\frac{1}{3} x-7\) \(y+3 x=1\)

Short Answer

Expert verified
The lines are perpendicular.

Step by step solution

01

- Write Both Equations in Slope-Intercept Form

The slope-intercept form of a line is given by the equation: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. The first line is already in this form: \[ y = \frac{1}{3} x - 7 \]The second line equation \( y + 3x = 1 \) needs to be rearranged to find the slope and y-intercept: Subtract \(3x\) from both sides to get:\[ y = -3x + 1 \]
02

- Identify Slopes of Both Lines

From the slope-intercept form: First line: \( y = \frac{1}{3} x - 7 \), the slope \( m_1 = \frac{1}{3} \) Second line: \( y = -3x + 1 \), the slope \( m_2 = -3 \)
03

- Determine Relationship Between Slopes

To determine if lines are parallel, perpendicular, or neither: Two lines are parallel if their slopes are equal: \( m_1 = m_2 \) Two lines are perpendicular if the product of their slopes is \(-1\): \( m_1 \times m_2 = -1 \)Otherwise, the lines are neither. In this case: \( \frac{1}{3} \times -3 = -1 \) which satisfies the condition for perpendicular lines.

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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 line's equation is a way to describe a line algebraically. It is written as: \[ y = mx + b \] In this format, \( y \) represents the y-coordinate of any point on the line, \( x \) represents the x-coordinate, \( m \) is the slope of the line, and \( b \) is the y-intercept, which is where the line crosses the y-axis. This form is very useful for graphing lines and understanding their behavior. For example, a line in slope-intercept form makes it easy to see how steep the line is and where it starts on the y-axis. The exercise involved converting both lines to this form, which made it simpler to identify their slopes and compare them.
slope
The slope of a line measures its steepness or incline. Mathematically, it is defined as the ratio of the vertical change to the horizontal change between two points on the line: \[ m = \frac{{\text{{change in y}}}}{{\text{{change in x}}}} \] The slope can tell us a lot about the line:
  • A positive slope means the line goes up as you move from left to right.
  • A negative slope means the line goes down as you move from left to right.
  • A slope of zero means the line is horizontal.
  • An undefined slope (division by zero) means the line is vertical.
In the example given, the first line had a slope \( m_1 = \frac{1}{3} \), indicating it rises gradually. The second line had a slope \( m_2 = -3 \), meaning it declines steeply.
line equations
Line equations help us describe the relationship between x and y coordinates on a Cartesian plane. Common forms of line equations include:
  • Slope-intercept form: \[ y = mx + b \]

  • Standard form: \[ Ax + By = C \]

  • Point-slope form: \[ y - y_1 = m(x - x_1) \]
The slope-intercept form is often the most intuitive and straightforward to use, especially for quick graphing. To solve our exercise, we converted both line equations into slope-intercept form. This made it easy to compare their slopes directly and determine their relationships (parallel, perpendicular, or neither). By doing so, we found that these lines were perpendicular because the product of their slopes was \( -1 \).
algebra
Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It helps us represent real-world relationships using equations and formulas. Understanding how to manipulate these equations is crucial for solving various problems. In the given exercise, algebra was used to:
  • Convert the equation \( y + 3x = 1 \) into slope-intercept form \( y = -3x + 1 \)
  • Identify and compare the slopes of two lines
  • Determine their geometric relationship (parallel, perpendicular, or neither)
This exercise demonstrated how algebra can be applied to real-world problems involving line equations and geometric interpretations. By understanding algebra, you can solve problems systematically and understand how different mathematical concepts connect. It's an essential skill for tackling more complex mathematical challenges.

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Most popular questions from this chapter

Solve each system using the substitution method. $$ \begin{aligned} &2 y+x=9\\\ &x=3 y-3 \end{aligned} $$

Solve each system using the elimination method. $$ \begin{array}{l} {5 r-3 s=24} \\ {3 r+5 s=28} \end{array} $$

Solve each system graphically. $$ \begin{aligned} &x+y=7\\\ &x-y=1 \end{aligned} $$

Translate each problem to an equation. Do not solve. It takes 42 gal of water to produce 1 lb of broccoli. This is twice the amount of water used to produce 1 16 of lettuce. How many gallons of water does it take to produce 1 lb of lettuce?

Translate each problem to an equation. Do not solve. Three times what number is \(348 ?\)
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