/*! 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 55 Write an equation of the line th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 55

Write an equation of the line that passes through the points. $$(1,-4),(2,-1)$$

Short Answer

Expert verified
The equation of the line that passes through the points (1, -4) and (2, -1) is \(y = 3x - 7\).

Step by step solution

01

Calculate the slope

Use the formula for the slope which is \(m = \frac{(y2 - y1)}{(x2 - x1)}\). Substitute the given coordinates into the formula, so it becomes \(m = \frac{(-1 - (-4))}{(2 - 1)}\) which simplifies to \(m = 3\).
02

Write the equation

Use the point-slope form \(y - y1 = m(x - x1)\) to write the equation. Substitute m = 3 and use the point (1,-4) for (x1, y1) to get the equation \(y - (-4) = 3(x - 1)\). This simplifies to \(y + 4 = 3x - 3\). After rearranging the numbers, the final equation is \(y = 3x - 7\).

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 Calculation
Calculating the slope is an essential first step in understanding linear equations. The slope of a line measures its steepness and direction. You can determine the slope by using two points on the line. Here's the formula you use:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
In this formula, \( m \) stands for the slope, \((x_1, y_1)\) and \((x_2, y_2)\) represent the coordinates of the two points. Insert the values from the points given in the exercise:
  • \( x_1 = 1, y_1 = -4 \)
  • \( x_2 = 2, y_2 = -1 \)
Substitute these into the formula:
  • \( m = \frac{-1 - (-4)}{2 - 1} \)
  • \( m = \frac{3}{1} \)
Thus, the slope \( m \) is 3, meaning for every unit you move the line horizontally, it moves three units vertically.
Point-Slope Form
The point-slope form of a linear equation is a crucial tool when you need to find the equation of a line. It's particularly helpful if you know the slope and a single point on the line. The general representation of this form is:
  • \( y - y_1 = m(x - x_1) \)
In this equation:
  • \((x_1, y_1)\) are the coordinates of the known point
  • \(m\) is the slope of the line
Using the point (1, -4) and the slope \( m = 3 \), plug these values into the formula:
  • \( y - (-4) = 3(x - 1) \)
  • This simplifies to \( y + 4 = 3x - 3 \)
Now, you have a point-slope equation, which describes the relationship between \( y \) and \( x \) based on the slope and the known point.
Equation of a Line
Once you find the slope and understand the point-slope form, you can convert this into the standard equation of a line, which usually appears as \(y = mx + b\). The goal here is to express the equation in a more conventional form that is easier to use in various applications.From the point-slope form we derived:
  • \( y + 4 = 3x - 3 \)
You rearrange this to the standard form:
  • Subtract 4 from both sides: \( y = 3x - 3 - 4 \)
  • This simplifies to \( y = 3x - 7 \)
Now, \(y = 3x - 7\) represents the line's equation in its slope-intercept form, where 3 is the slope, and -7 is the y-intercept. This expression shows how \(y\) changes with \(x\), and serves as a tool for graphing or analyzing the line's behavior.

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

\(g(x)=0.75 x^2-5\)

\(f(x)=0.5 x^2+x-3\)

\(y=-\frac{5}{2} x^2-4 x-1\)

THOUGHT PROVOKING A jump on a pogo stick with a conventional spring can be modeled by f(x) = ?0.5(x ? 6)2 + 18, where x is the horizontal distance (in inches) and f(x) is the vertical distance (in inches). Write at least one transformation of the function and provide a possible reason for your transformation.

ANALYZING EQUATIONS The graph of which function has the same axis of symmetry as the graph of \(y=x^2+2 x+2 ?\) (A) \(y=2 x^2+2 x+2\) (B) \(y=-3 x^2-6 x+2\) (C) \(y=x^2-2 x+2\) (D) \(y=-5 x^2+10 x+2\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and 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.