/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 For exercises 1-8, find the slop... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 7

For exercises 1-8, find the slope of the line that passes through the given points. $$ \left(\frac{3}{8},-\frac{1}{2}\right)\left(-\frac{5}{8},-\frac{5}{2}\right) $$

Short Answer

Expert verified
The slope is 2.

Step by step solution

01

- Understand the Slope Formula

The slope of a line passing through two points \(x_1, y_1\) and \(x_2, y_2\) is given by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
02

- Identify Coordinates

Identify the coordinates \(x_1, y_1\) and \(x_2, y_2\) from the given points. Here, \(x_1 = \frac{3}{8}\), \(y_1 = -\frac{1}{2}\), \(x_2 = -\frac{5}{8}\), and \(y_2 = -\frac{5}{2}\).
03

- Subtract y-coordinates

Subtract the y-coordinates: \( y_2 - y_1 = -\frac{5}{2} - \left( -\frac{1}{2} \right) = -\frac{5}{2} + \frac{1}{2} = -\frac{4}{2} = -2 \).
04

- Subtract x-coordinates

Subtract the x-coordinates: \( x_2 - x_1 = -\frac{5}{8} - \left( \frac{3}{8} \right) = -\frac{5}{8} - \frac{3}{8} = -\frac{8}{8} = -1 \).
05

- Divide and Simplify

Divide the results from steps 3 and 4 to find the slope: \( m = \frac{-2}{-1} = 2 \). Hence, the slope of the line is 2.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

slope formula
To find the slope of a line, you need to understand and use the slope formula. The slope formula helps you determine the steepness or incline of a line that passes through two points. The formula is:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
In this formula, \( m \) represents the slope, and \( x_1, y_1 \) and \( x_2, y_2 \) are the coordinates of the two points through which the line passes. By applying this formula, you can easily calculate the slope. This formula is fundamental in coordinate geometry as it helps describe how one variable changes in relation to another.
coordinate geometry
Coordinate geometry, also known as analytic geometry, allows us to represent geometric figures and analyze their properties using a coordinate system. Here, we work with the Cartesian plane which is defined by the x-axis and y-axis. Every point on this plane can be represented by a pair of coordinates, \( (x, y) \). To solve the slope exercise, we use the coordinates of the given points. In our example, the coordinates are:
  • \( \frac{3}{8}, -\frac{1}{2} \) and \( -\frac{5}{8}, -\frac{5}{2} \)
By using these coordinates in our calculations, we follow a systematic way of solving geometric problems.
subtraction of fractions
When dealing with the slope formula, you'll often need to subtract fractions. For example, to find the change in y-coordinates, we perform the subtraction:
  • \( y_2 - y_1 = -\frac{5}{2} - (-\frac{1}{2}) = -\frac{5}{2} + \frac{1}{2} \)
Here we use the rule that subtracting a negative fraction is the same as adding its positive counterpart. The steps are as follows:
  • First, change the subtraction to addition where necessary.
  • Find a common denominator if needed.
  • Perform the operation on both the numerators.
In our case, \( -\frac{5}{2} + \frac{1}{2} = -\frac{4}{2} = -2 \). This operation is crucial for finding the correct numerator in the slope formula.
simplifying fractions
Simplifying fractions is an essential skill when working with the slope formula. Once you subtract the coordinates, you may need to simplify the resulting fraction. For example, let's look at the calculation of the change in x-coordinates:
  • \( x_2 - x_1 = -\frac{5}{8} - \frac{3}{8} \)
  • \( -\frac{5}{8} - \frac{3}{8} = -\frac{8}{8} = -1 \)
The fraction \( -\frac{8}{8} \) simplifies to -1. Simplifying fractions involves reducing them to their simplest form by dividing both the numerator and the denominator by their greatest common divisor. After simplifying, the final step is:
  • \( m = \frac{-2}{-1} \)
Simplifying \( \frac{-2}{-1} \) gives us the slope of 2. Always double-check your steps to ensure accuracy.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The relationship of the taxable value of a property, \(x\), and the annual property tax, \(y\), is a direct variation. When the taxable value of a property is \(\$ 250,000\), the annual property tax bill is \(\$ 5375\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this relationship. c. Find the taxable value of a property with an annual property tax bill of \(\$ 8062.50\). d. Find the tax owed for a property with an assessed value of \(\$ 185,000\). Round to the nearest whole number. e. What does \(k\) represent in this equation?

The rulebook of the U.S. Lawn Mower Racing Association describes how to award points. 100 points for registration 100 points for starting a race 100 points for finishing a race 300 points for first place 250 points for second place 200 points for third place 150 points for fourth place 100 points for fifth place Source: www.letsmow.com A lawn mower racer registered for a day of racing. She started and completed three races. She placed fourth in the first race, third in the second race, and first in the final race. Find the total number of points she earned.

When the top of a cone is removed, the formula for the volume of the remaining cone (the frustrum) is \(V=\frac{1}{3} \pi\left(R^{2}+R r+r^{2}\right) h\), where \(r\) is the radius of the circle at the top of the frustrum and \(R\) is the radius of the circle at the bottom of the frustrum. In 1856, an American army officer, Henry Hopkins Sibley, invented and received a patent for the design of a conical tent that could sleep 12 soldiers. (The apex is the diameter of the top of the frustrum.) Find the volume of the tent in cubic feet. Use \(\pi \approx 3.14\). Round to the nearest whole number. Be it known that I, H.H. Sibley, United States Army, have invented a new and improved Conical Tent ... the tent is in shape the frustrum of a cone; the base 18 feet; the height 12 feet; the apex 1 foot 6 inches [1.5 ft]. (Source: patimg1.uspto.gov)

For exercises \(67-82\), use the five steps and a proportion. In 2010 , there were \(14.9\) cases of syphilis per 100,000 Americans with a total of 45,834 cases of syphilis. Find the population of Americans used to create this ratio. Round to the nearest hundred. (Source: www.cdc.gov, 2011)

When a car travels a fixed distance, the relationship between the speed of the car, \(x\), and the time it travels, \(y\), is an inverse variation. When the speed is \(\frac{48 \mathrm{mi}}{1 \mathrm{hr}}\), the time is \(0.75 \mathrm{hr}\). a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the time in hours to travel this distance at a speed of \(\frac{80 \mathrm{mi}}{1 \mathrm{hr}}\). d. Change the time in part \(\mathrm{c}\) to minutes.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.