/*! 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 52 Find the slope of the line passi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 52

Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. \((a-b, c)\) and \((a, a+c)\)

Short Answer

Expert verified
The slope of the line through the points \( (a-b, c) \) and \( (a, a+c) \) is \(b\), and the line rises from left to right.

Step by step solution

01

Identifying the coordinates

The coordinates of the two points are \((a-b, c)\) and \((a, a+c)\). So, \(x1 = a - b\), \(y1 = c\), \(x2 = a\), and \(y2 = a + c\).
02

Calculating the slope (m)

Substituting the given points into the slope formula: \( m = (y2 - y1) / (x2 - x1) = ( (a + c) - c ) / ( a - (a - b) ) = b\).
03

Analyzing the Slope

Since the computed slope (b) is positive, and assumed to be a positive real number, the line rises from left to right. If the slope were negative, the line would fall. A slope of 0 means the line is horizontal, and an undefined slope means the line is vertical.

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.

Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a fascinating way of representing geometric figures using algebraic equations. When we work with points on a plane, we often use coordinates, which are pairs of numbers that describe the location of each point. These coordinates are based on the Cartesian coordinate system, consisting of an x-axis (horizontal) and y-axis (vertical). Each point is defined by an ordered pair
  • The first number represents its position along the x-axis.
  • The second number represents its position along the y-axis.

In our given problem, two points are provided,
  • one at a-b
Equation of a Line
The equation of a line is a mathematical representation that describes a straight line's path on the plane. The most common form is the slope-intercept form, given by \[ y = mx + b \]. Here,
  • \(m\) is the slope, illustrating how steep the line is.
  • \(b\) is the y-intercept, showing where it intersects the y-axis.

To find the slope, we use the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
In our exercise, we calculated the slope as \( m = b \), given the points
  • ( a-b , c )
  • and ( a, a+c )

By confirming the slope as \(b\), we can determine the line鈥檚 behavior, using it in our equation to define its path across the coordinate system. This provides a fundamental tool in coordinate geometry, allowing us to predict the position of points along the line.
Positive Slope
A positive slope in the context of a line indicates that the line ascends as you move from left to right across the graph. This quality of the line is directly linked to the value of the slope, which is a representation of how rapidly the line rises.

The computation given in the solution revealed a slope of \( b \), implying:
  • As the slope is positive, the line rises with the increasing x values.
  • The rate of ascent is dictated by the magnitude of \( b \).

Understanding whether a slope is positive is crucial in applications like economics to signal trends, or physics for motion analysis, as it helps to determine the relationship between variables and predict future values.
Line Analysis
Line analysis involves a detailed study of the line's properties, determining important factors such as slope, direction, and graph position. In this problem, after calculating the slope, the next step is to interpret these findings:
  • A positive slope means that the graph of the line will incline upwards as it moves from left to right.
  • If the calculated slope had been zero, this would indicate a horizontal line, reflecting no change in y-value as x changes.
  • An undefined slope is associated with a vertical line, where x-value remains constant while y changes.

In our exercise, because the slope is positive (\(b\)), it tells us that the line rises, conveying an increasing relationship between the x and y coordinates. By analyzing slopes effectively, you gain a deeper insight into how lines behave and how different scenarios appear graphically.

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

Use the directions for Exercises 5-8 to solve Exercises 9-12. Objective Function $$ z=5 x-2 y $$ Constraints $$ \left\\{\begin{array}{l} 0 \leq x \leq 5 \\ 0 \leq y \leq 3 \\ x+y \geq 2 \end{array}\right. $$

On June 24,1948 , the former Soviet Union blocked all land and water routes through East Germany to Berlin. A gigantic airlift was organized using American and British planes to bring food, clothing, and other supplies to the more than 2 million people in West Berlin. The cargo capacity was 30,000 cubic feet for an American plane and 20,000 cubic feet for a British plane. To break the Soviet blockade, the Western Allies had to maximize cargo capacity, but were subject to the following restrictions: \- No more than 44 planes could be used. \- The larger American planes required 16 personnel per flight, double that of the requirement for the British planes. The total number of personnel available could not exceed 512 . \- The cost of an American flight was \(\$ 9000\) and the cost of a British flight was \(\$ 5000\). Total weekly costs could not exceed \(\$ 300,000\). Find the number of American and British planes that were used to maximize cargo capacity.

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \geq 2 \\ y \leq 3\end{array}\right.\)

a. Determine if the parabola whose equation is given opens upward or downward. b. Find the vertex. c. Find the \(x\)-intercepts. d. Find the y-intercept. e. Use (a)-(d) to graph the quadratic function. \(y=-x^{2}+4 x-3\)

You are about to take a test that contains computation problems worth 6 points each and word problems worth 10 points each. You can do a computation problem in 2 minutes and a word problem in 4 minutes. You have 40 minutes to take the test and may answer no more than 12 problems. Assuming you answer all the problems attempted correctly, how many of each type of problem must you answer to maximize your score? What is the maximum score?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

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.