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Chapter 1: Problem 52

Determine whether the statement is true or false. The lines \(y=2 x-3\) and \(y=-2 x-3\) are perpendicular.

Short Answer

Expert verified
False

Step by step solution

01

Identify the slopes of the lines

First, identify the slopes of the given lines. For the line equation in the form of 饾懄 = 饾憵饾懃 + 饾憦, 饾憵 is the slope. For the line equation \(y = 2x - 3\), the slope \(m_1\) is 2. For the line equation \(y = -2x - 3\), the slope \(m_2\) is -2.
02

Check the criteria for perpendicular lines

Two lines are perpendicular if and only if the product of their slopes is -1. Mathematically, this means \(m_1 \times m_2 = -1\).
03

Calculate the product of the slopes

Now, calculate the product of the two slopes. \(m_1 = 2\) and \(m_2 = -2\), so the product \(m_1 \times m_2 = 2 \times -2 = -4\).
04

Determine if the product meets the perpendicular criteria

Since the product of the slopes, -4, does not equal -1, the lines are not perpendicular.

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Key Concepts

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

slope
The concept of slope is crucial in understanding the behavior of lines on a graph.
In the formula of a straight line, like in the format of 饾懄 = 饾憵饾懃 + 饾憦, the term '饾憵' represents the slope.
The slope dictates how steep the line is.
  • If the slope is positive, the line rises from left to right.
  • If the slope is negative, the line falls from left to right.
  • A zero slope makes a horizontal line, while an undefined slope creates a vertical line.
In our example, the slopes are found to be 2 and -2, indicating that the lines are neither horizontal nor vertical but have different inclinations.
linear equations
Linear equations are equations that make straight lines when plotted on a graph.
They commonly appear in the form 饾懄 = 饾憵饾懃 + 饾憦, where
  • '饾憵' is the slope of the line
  • '饾憦' is the y-intercept, the point where the line crosses the y-axis.
For our lines, 饾懄 = 2饾懃 - 3 and 饾懄 = -2饾懃 - 3, we identified:
  • The slopes, 饾憵1= 2 and 饾憵2= -2
  • The y-intercepts, both at -3, which means they cross the y-axis at the same point.
Understanding linear equations is key to solving many geometric problems.
geometric relationships
Geometric relationships help us understand how different shapes and lines interact.
One important relationship is perpendicularity. Two lines are perpendicular if their slopes multiply to -1.
In mathematical terms, if line A has slope 饾憵1 and line B has slope 饾憵2, then:
  • They are perpendicular if and only if 饾憵1*饾憵2 = -1.
In our example, we have slopes 2 and -2. When we multiply them together, we get -4, not -1.
Therefore, the lines 饾懄 = 2饾懃 - 3 and 饾懄 = -2饾懃 - 3 are not perpendicular.
Understanding these relationships can make it easier to solve many geometry problems.

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Most popular questions from this chapter

The average worldwide daily calorie intake per person has increased from 2200 to 2800 calories since the early 1960 s. The average daily calorie intake per person in the United States is \(3688 .\) This is about \(86.4 \%\) more than the average daily calorie intake per person in Haiti. (Sources: UN Food and Agriculture Organization; World Health Organization) Find the average daily calorie intake per person in Haiti.

Find the distance between the pair of points. Give an exact answer and, where appropriate, an approximation to three decimal places. $$(-11,-8) \text { and }(1,-13)$$

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Find the distance between the pair of points. Give an exact answer and, where appropriate, an approximation to three decimal places. $$\left(-8, \frac{7}{11}\right) \text { and }\left(8, \frac{7}{11}\right)$$
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