/*! 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 15 Write an equation for each line ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 15

Write an equation for each line in the indicated form. Write the equation of the line passing through the points (-4,4) and (0,-4) in slope-intercept form.

Short Answer

Expert verified
The equation is \( y = -2x - 4 \).

Step by step solution

01

Determine the Slope

To find the equation of a line, we first need to calculate the slope using the two given points: \((-4, 4)\) and \((0, -4)\). The formula for the slope \(m\) is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Substituting the points into the formula gives us: \[ m = \frac{-4 - 4}{0 - (-4)} = \frac{-8}{4} = -2 \]. The slope of the line is \(-2\).
02

Use Point-Slope Form

We now use the point-slope form of a line, which is \( y - y_1 = m(x - x_1) \), with one of the points, let's use \((0, -4)\), and the slope \( m = -2 \). Substituting these values in, we get: \[ y - (-4) = -2(x - 0) \], which simplifies to \[ y + 4 = -2x \].
03

Convert to Slope-Intercept Form

The slope-intercept form of a line is \( y = mx + b \). From \( y + 4 = -2x \), isolate \( y \) to get the line equation in slope-intercept form: \[ y = -2x - 4 \].
04

Verify the Line Equation

To ensure the equation \( y = -2x - 4 \) is correct, substitute the original points \((-4, 4)\) and \((0, -4)\) into the equation and check if the left-hand side equals the right-hand side. For \(x = -4\), \(y = 4\), the calculation is \(4 = -2(-4) - 4\) which simplifies to \(4 = 8 - 4\), true. For \(x = 0\), \(y = -4\), \(-4 = -2(0) - 4\) simplifies to \(-4 = -4\), also true. The equation is verified.

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
In coordinate geometry, the slope of a line is a measure of how steep the line is. It tells us how much one variable changes for a unit change in another. To calculate the slope (often represented as \(m\)) between two points, we use the formula:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
This formula helps us determine the rate at which \(y\) is changing with respect to \(x\). In the problem, the points \((-4, 4)\) and \((0, -4)\) were given, and using the slope formula:
  • \( m = \frac{-4 - 4}{0 - (-4)} = \frac{-8}{4} = -2 \)
The resultant slope is \(-2\), indicating that for every unit increase in \(x\), \(y\) decreases by 2 units. This negative slope confirms the line is declining as it moves from left to right.
Point-Slope Form
Once the slope is calculated, the point-slope form of a line can be used to quickly write an equation for the line. The standard form is:
  • \( y - y_1 = m(x - x_1) \)
It utilizes a known point \((x_1, y_1)\) on the line and the slope \(m\). This is particularly useful when you have one point and the slope already figured out. For the exercise, using the point \((0, -4)\) and the slope \(-2\) we substitute in the format:
  • \( y - (-4) = -2(x - 0) \)
Simplifying gives us \( y + 4 = -2x \). This form is very flexible for situations where you might need to write the equation a line based on partial information.
Line Equation
The line equation is essentially the algebraic representation of a straight line in a plane. To put a line equation in the more familiar slope-intercept form, we can convert it from other forms. Slope-intercept form is given by:
  • \( y = mx + b \)
where \(m\) is the slope and \(b\) is the y-intercept. From our previous derivation in the exercise of \( y + 4 = -2x \), isolating \(y\) gives:
  • \( y = -2x - 4 \)
This is the completed line equation in slope-intercept form, indicating that the line crosses the y-axis at \(y = -4\), with a slope of \(-2\). Such conversion makes graphing the line a lot simpler and is widely used in mathematical analyses.
Coordinate Geometry
Coordinate geometry, or analytic geometry, bridges algebra and geometry by representing geometric figures with equations. In this context, working with a line's equation shows how we can utilize its algebraic form for various calculations:
  • Finding intersection points
  • Determining parallelism and perpendicularity
  • Exploring distances between points on the line
In the exercise, coordinate geometry allows us to verify the correctness of our line equation. By plugging in the original points \((-4, 4)\) and \((0, -4)\) into the equation \( y = -2x - 4 \), we check for consistency:
  • For \((0, -4)\), \(-4 = -2(0) - 4\) which is true
  • For \((-4, 4)\), \(4 = -2(-4) - 4\) which is true
Coordinate geometry thus provides a robust framework to ensure that equations are constructed and manipulated accurately.

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

Answer each question about the properties of the given line(s). Consider the linear function \(g(x)=-4 x+5 .\) What is the slope of the function when \(x=4 ?\)

In exercises \(12-16,\) write each function in the form \(c(x+\) \(a)^{2}+b\) and identify the values of \(a, b\), and \(c\). $$ y(x)=9 x^{2}+18 x+4 $$

In exercises \(5-11,\) write each function in the form \((x+a)^{2}+b\) and identify the values of \(a\) and \(b\). $$ g(x)=x^{2}+20 x+40 $$

After completing the square you get that \(g(t)=(t+2)^{2}-6\). What are the values of \(a\) and \(b\) if your goal form is \((x+a)^{2}+\) b?

Answer each question about the properties of the given line(s). Determine the \(x\) -intercept of the line \(y=4 x-8\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision 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.