/*! 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 47 In Exercises 41-50, use the poin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 47

In Exercises 41-50, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) \( (-5, 4) \), \( m = 2 \)

Short Answer

Expert verified
The three additional points on the line are \((0, 14), (-2, 10), (2, 18)\).

Step by step solution

01

Find the y-intercept

Substitute the known point \((-5, 4)\) and the slope \(2\) into the equation \(y = mx + b\). This gives you a new equation \(4 = 2*(-5) + b\), which simplifies to \(b = 14\).
02

Find Additional Points

Using the equation \( y = 2x + 14 \), select arbitrary values for \(x\) and solve for \(y\). Consider \(x = 0, -2, 2\), the solutions are \( (0,14), \, (-2,10), \, (2,18) \) respectively.
03

Confirm Points

Ensure that all points lie on the line by substituting the x and y values into the line equation \( y = 2x + 14\) and ensuring that both sides hold equal.

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-Intercept Form
When dealing with linear equations in algebra, one of the most common forms you will encounter is the slope-intercept form. This form makes understanding and graphing lines straightforward. The general equation is represented as \( y = mx + b \). Here, \( m \) stands for the slope of the line, indicating how steep the line is. The term \( b \) denotes the y-intercept, which is the point where the line crosses the y-axis.

Understanding the components:
  • **Slope (\( m \))**: This tells us the direction and steepness of the line. A positive slope means the line rises as it moves from left to right. A negative slope indicates the line falls.
  • **Y-intercept (\( b \))**: Found by setting \( x = 0 \), it's the point where your line will cross the y-axis.

The ease of knowing a line's slope and y-intercept means you can sketch the line quickly, making it invaluable in practical applications like this exercise.
Finding Points on a Line
Once the slope-intercept form of the equation is established, finding additional points on a line becomes an easy task. Let's see how to perform this:
  • Start with the slope-intercept form of the equation. In this exercise, the equation is \( y = 2x + 14 \).
  • Select different values for \( x \). This choice is arbitrary, but common selections are integers for simplicity.
  • Plug each \( x \) value into the equation to solve for \( y \). Each \( (x, y) \) pair is a point on the line.
To illustrate: For \( x = 0 \), inserting into the equation gives \( y = 2(0) + 14 = 14 \), hence the point \( (0, 14) \). By repeating this for multiple \( x \) values, like \( x = -2 \) or \( x = 2 \), you can generate as many points as needed.
Verifying the accuracy of these points involves substituting them back into the equation. If both sides match, the points are confirmed to be on the line.
Coordinate Geometry
In coordinate geometry, we often describe the location and properties of figures using a coordinate system. This system lets us place each mathematical point as an ordered pair \((x, y)\) on a 2D plane.
Coordinate geometry helps visualization by providing a clear visual representation of algebraic equations on a plane.
  • Using concepts like the slope-intercept form, we can model lines within this plane and determine intersections, parallelism, and various other relationships.
  • Finding points on a line extends beyond mere numbers; it's about understanding how each point relates to the equation and the overall geometric figure.

Connections in coordinate geometry make it easier to transition from abstract algebraic expressions to tangible diagrams, which is why mastering these concepts is fundamental to many scientific fields.

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

DIRECT VARIATION In Exercises 35-38, assume that is \(y\) directly proportional to \(x\). Use the given \(x\)-value and \(y\)-value to find a linear model that relates \(y\) and \(x\). \(x = 6\), \(y = 580\)

In Exercises 63-76, determine whether the function has an inverse function. If it does, find the inverse function. \( f(x) = \left\\{ \begin{array}{ll} x+3, & \mbox{ \) x < 0 \(} \\ 6-x, & \mbox{ \) x \geq 0 \(} \end{array} \right.\)

In Exercises 49-62, (a) find the inverse function of \(f\) (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). \(f{x} = \frac{6x+4}{4x+5}\)

SPORTS The winning times (in minutes) in the women's 400-meter freestyle swimming event in the Olympics from 1948 through 2008 are given by the following ordered pairs. \((1948, 5.30)\) \((1952, 5.20)\) \((1956, 4.91)\) \((1960, 4.84)\) \((1964, 4.72)\) \((1968, 4.53)\) \((1972, 4.32)\) \((1976, 4.16)\) \((1980, 4.15)\) \((1984, 4.12)\) \((1988, 4.06)\) \((1992, 4.12)\) \((1996, 4.12)\) \((2000, 4.10)\) \((2004, 4.09)\) \((2008, 4.05)\) A linear model that approximates the data is \(y = -0.020t + 5.00\), where \(y\) represents the winning time (in minutes) and \(t=0\) represents 1950. Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? Does it appear that another type of model may be a better fit? Explain. (Source: International Olympic Committee)

In Exercises 63-76, determine whether the function has an inverse function. If it does, find the inverse function. \(f(x) = \sqrt{2x+3}\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

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.