/*! 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 43 Find an equation of the line tha... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 43

Find an equation of the line that passes through the given point and has the indicated slope \(m .\) Sketch the line. $$(0,-2), \quad m=3$$

Short Answer

Expert verified
The equation of the line passing through (0, -2) with a slope of 3 is \(y=3x-2\)

Step by step solution

01

Identify the given slope and point

The slope \(m\) is given as 3 and the point is (0, -2).
02

Compute the y-intercept from the given slope and point

To find the y-intercept, we plug in the slope and the provided points to the slope-intercept formula, \(y = mx + b\). \nWe get: \(-2 = 3*0 + b\). \nSolving for \(b\), we get \(b = -2\)
03

Formulate the equation of the line

Substitute the slope and y-intercept into the slope-intercept formula. \nThe equation of line thus becomes \(y = 3x - 2\)
04

Sketch the Line

To sketch the line, first draw an x-y plane. Then plot the y-intercept, which is -2. Starting from the y-intercept, use the slope to find the next point. Here, the slope is \3, meaning for every 1 unit positive change in x, y increases by 3 units. Using these points, draw the line.

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
The slope-intercept form is one of the most common ways to represent the equation of a line. It is expressed as \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept of the line. This formula offers a straightforward way to graph a line and understand its properties. The slope, \(m\), indicates the steepness and direction of the line:
  • A positive slope means the line inclines upwards.
  • A negative slope suggests the line declines downwards.
For example, in the equation \(y = 3x - 2\), the slope \(m\) is 3, meaning the line rises 3 units for every unit it moves to the right. The y-intercept \(b\) is -2, showing where the line crosses the y-axis. This makes it easy to quickly draw the line or determine how the line behaves across different points on the graph.
Point-Slope Form
The point-slope form is especially handy when you know a point on the line and the slope. It is given by the formula \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a specific point on the line, and \(m\) is the slope. This form emphasizes how a particular point and the line's slope are enough to describe the line fully.
  • The benefit of this format is that it simplifies the process of formulating a line equation from practical data points.
  • It is particularly useful when graphical plotting isn't possible initially, but precise points are known.
In our original exercise, if we begin with the point (0, -2) and the slope \(m = 3\), the equation can be formatted as \(y + 2 = 3(x - 0)\). From here, we can transition to slope-intercept form and easily understand how the line will appear.
Graphing Linear Equations
Graphing linear equations is the visual representation of lines on a coordinate plane. It is a crucial skill that helps us visually interpret relationships described by formulas like the slope-intercept or point-slope forms. To graph an equation such as \(y = 3x - 2\):
  • Start by plotting the y-intercept, which is the point \((0, -2)\) on the y-axis.
  • From the y-intercept, apply the slope \(m = 3\). This means moving 3 units up for every 1 unit right from the y-intercept to find another point, e.g., the point \((1, 1)\).
  • Plot additional points following this rise-over-run pattern if needed.
  • Finally, draw a straight line through all these points using a ruler for accuracy.
Graphing allows us to quickly see important features, like where two lines might intersect or how they parallel each other, and makes realizing the line's extension across the plane an easy task.

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

Evaluate the function for the indicated values. \(h(x)=[x+3]\) (a) \(h(-2)\) (b) \(h\left(\frac{1}{2}\right)\) (c) \(h(4.2)\) (d) \(h(-21.6)\)

Determine whether the statement is true or false. Justify your answer. Every relation is a function.

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(z\) varies directly as the square of \(x\) and inversely as \(y\) \((z=6 \text { when } x=6 \text { and } y=4 .)\)

Determine whether the statement is true or false. Justify your answer. The set of ordered pairs \(\\{(-8,-2),(-6,0),(-4,0)\) \((-2,2),(0,4),(2,-2)\\}\) represents a function.

The cost per unit in the production of an MP3 player is 60 dollars. The manufacturer charges 90 dollars per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by 0.15 dollars per MP3 player for each unit ordered in excess of 100 (for example, there would be a charge of 87 dollars per MP3 player for an order size of 120 ). (a) The table shows the profits \(P\) (in dollars) for various numbers of units ordered, \(x .\) Use the table to estimate the maximum profit. $$\begin{array}{|l|c|c|c|c|c|}\hline \text { Units, } x & 130 & 140 & 150 & 160 & 170 \\\\\hline \text { Profit, } P & 3315 & 3360 & 3375 & 3360 & 3315 \\\\\hline\end{array}$$ (b) Plot the points \((x, P)\) from the table in part (a). Does the relation defined by the ordered pairs represent \(P\) as a function of \(x ?\) (c) Given that \(P\) is a function of \(x,\) write the function and determine its domain. (Note: \(P=R-C\) where \(R\) is revenue and \(C\) is cost.)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

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.