/*! 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 18 Use the given conditions to writ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 18

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-2,\) passing through \((0,-3)\)

Short Answer

Expert verified
The point-slope form of the line is \(y + 3 = -2x\) and the slope-intercept form is \(y = -2x -3\).

Step by step solution

01

Formulate the Point-Slope Form

The point-slope form of the line is \(y - y_1 = m(x - x_1)\) where \(m\) is the slope and \((x_1, y_1)\) is the point that the line passes through. Substituting the given values: \(m = -2\) and the point \((0, -3)\) into the equation, we get: \(y - (-3) = -2 * [x-0]\). Which simplifies to \(y + 3 = -2x\).
02

Convert the Point-Slope Form into Slope-Intercept Form

The slope-intercept form of the line is \(y = mx + c\) which, by rearranging the equation from point-slope form \((y + 3 = -2x)\), results in \(y = -2x - 3\).

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
In mathematics, the slope-intercept form is used to express the equation of a straight line. This form is represented as: \(y = mx + c\), where:
  • \(y\) is the dependent variable representing the y-coordinate of a point on the graph.
  • \(m\) is the slope of the line, indicating its steepness and direction.
  • \(x\) is the independent variable representing the x-coordinate of a point.
  • \(c\) is the y-intercept, the point where the line crosses the y-axis.
This form is particularly useful because it readily provides the slope and y-intercept, allowing one to quickly graph the line or understand its behavior. The conversion from point-slope form to slope-intercept form requires simply rearranging the equation, ensuring \(y\) is isolated. For example, from \(y + 3 = -2x\) in the point-slope form, it can be rearranged to \(y = -2x - 3\). Here, the slope \(m\) is \(-2\), and the y-intercept \(c\) is \(-3\).
Linear Equations
Linear equations describe straight lines in algebra. A key feature of these equations is their ability to represent constant relationships between two variables, usually x and y. The general form of a linear equation is \(ax + by = c\), but it can also be presented in slope-intercept form \(y = mx + c\).
  • They have a fixed slope (steepness) that never changes along the line.
  • Linear equations do not have exponents higher than 1 for the variables.
  • The graph of a linear equation is always a straight line.
In practical problems, linear equations help model relationships between quantities, such as speed and time or cost and items bought. Solving a linear equation involves finding the values of variables that make the equation true, often revealing their plotted line's path and intersection with axes.
Coordinates
Coordinates are essential in geometry and algebra as they provide a way to locate points on a graph. Typically expressed in the form \((x, y)\), coordinates contain:
  • The x-coordinate, which determines a specific position along the horizontal x-axis.
  • The y-coordinate, which determines a position along the vertical y-axis.
This coordinate pair allows for precise positioning of points in a two-dimensional space. For example, the point \((0, -3)\) signifies a location on the y-axis, indicating that this line passes through the y-axis at \(-3\). When working with linear equations like the point-slope form, understanding coordinates aids in plotting the initial point the line passes through, facilitating the graph's creation and analysis.
Algebraic Equations
Algebraic equations represent relationships between variables and constants, often involving addition, subtraction, multiplication, and division. These equations can be simple, like linear equations, or complex with higher-degree polynomials. In the context of this exercise:
  • The algebraic equation in point-slope form is \(y - y_1 = m(x - x_1)\).
  • Algebraic equations require substitution of known values to find unknowns.
  • They are manipulated through operations to solve for variables or simplify expressions.
Algebraic equations are the backbone of problems requiring solutions through manipulation of known and unknown quantities. By substituting the values \(m = -2\), \(x_1 = 0\), and \(y_1 = -3\) into the point-slope equation, we can derive a specific form illustrating the relationship between x and y for that line.

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

Then use the TRACE feature to trace along the line and find the coordinates of two points. Use these points to compute the line's slope. Check your result by using the coefficient of \(x\) in the line's equation. $$y=2 x+4$$

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)=\sqrt{x}+2 $$

During a particular year, the taxes owed, \(T(x),\) in dollars, filing separately with an adjusted gross income of \(x\) dollars is given by the piecewise function $$ T(x)=\left\\{\begin{array}{ll} 0.15 x & \text { if } 0 \leq x<17,900 \\ 0.28(x-17,900)+2685 & \text { if } 17,900 \leq x<43,250 \\ 0.31(x-43,250)+9783 & \text { if } x \geq 43,250 \end{array}\right. $$ In Exercises \(89-90,\) use this function to find and interpret each of the following. 90\. \(T(70,000)\)

You will be developing functions that model given conditions. You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. Write the total time, \(T,\) in hours, devoted to your outgoing and return trips as a function of your rate on the outgoing trip. Then find and interpret \(T(30) .\) Hint: Time traveled \(=\frac{\text { Distance traveled }}{\text { Rate of travel }}\)

Begin by graphing the standard cubic function, \(f(x)=x^{3} .\) Then use transformations of this graph to graph the given function. $$ h(x)=\frac{1}{4} x^{3} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.