/*! 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 435 Find the equation of a line cont... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 435

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (-2,8) and (-4,-6)

Short Answer

Expert verified
y = 7x + 22

Step by step solution

01

- Identify the Coordinates

Identify the coordinates of the given points. The given points are (-2, 8) and (-4, -6).
02

- Calculate the Slope

Use the formula for the slope of a line passing through two points: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the coordinates of the points (-2, 8) and (-4, -6): \[ m = \frac{-6 - 8}{-4 - (-2)} = \frac{-14}{-2} = 7 \] Hence, the slope (\text{m}) is 7.
03

- Use the Point-Slope Form

Use the point-slope form of the equation of a line:\[ y - y_1 = m(x - x_1) \] Choose one of the given points (-2, 8) and substitute it into the formula: \[ y - 8 = 7(x + 2) \]
04

- Simplify to Slope-Intercept Form

Simplify the equation into the slope-intercept form (\text{y = mx + b}): \[ y - 8 = 7(x + 2) \] Distribute the 7: \[ y - 8 = 7x + 14 \] Add 8 to both sides: \[ y = 7x + 22 \] Therefore, the equation of the line in slope-intercept form is y = 7x + 22.

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.

Equation of a Line
An equation of a line represents all the points that lie on that line. Typically, we use the slope-intercept form of the equation:y = mx + bHere:
  • y stands for the y-coordinate of any point on the line.
  • m represents the slope (steepness) of the line.
  • x is the x-coordinate of any point on the line.
  • b is the y-intercept, where the line crosses the y-axis.
Every straight line can be described with an equation like this, which makes it easy to predict y for any x.
Slope Calculation
The slope of a line measures its steepness. A positive slope means the line rises as it goes from left to right, while a negative slope means it falls. To find the slope between two points y_1 and y_2 with coordinates (x_1, y_1) and (x_2, y_2), we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In our example, the points are (-2, 8) and (-4, -6). Plugging in these coordinates, we get:\[ m = \frac{-6 - 8}{-4 - (-2)} = \frac{-14}{-2} = 7 \]Thus, the slope is 7, meaning the line rises steeply upwards.
Point-Slope Form
The point-slope form of a line's equation is useful for quickly forming the equation when you know a point and the slope. It looks like this: \[ y - y_1 = m(x - x_1) \]Here:
  • m is the slope.
  • (x_1, y_1) is a known point on the line.
Using one of our points (-2, 8) and the slope of 7, we substitute into the point-slope form:\[ y - 8 = 7(x + 2) \]This equation represents the same line but in a different format than the slope-intercept form.
Coordinate Geometry
Coordinate geometry, or analytic geometry, bridges algebra and geometry using coordinates. By plotting points, lines, and curves on an x-y plane, we can solve geometric problems algebraically.to understand why coordinate geometry is powerful:
  • It allows for precise plotting and measuring of geometric forms.
  • It simplifies complex problems into manageable algebraic equations.
  • You can visually see the relationship between equations and their graphical counterparts.
In our problem, the known points (-2, 8) and (-4, -6) define a line. By using coordinate geometry techniques, we derived the line's equation and analyzed its properties like slope.

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

Find the equation of each line. Write the equation in slope-intercept form. Containing the points (-3,-4) and (2,-5)

Determine whether each ordered pair is a solution to the inequality \(x+y>2\) : (a) (1,1) (b) (4,-3) (c) (0,0) (d) (-8,12) (e) (3,0)

Find the equation of each line. Write the equation in slope-intercept form. Parallel to the line \(4 x+3 y=6\), containing point(0,-3)

Find the equation of a line containing the given points. Write the equation in slope-intercept form. . (6,2) and (-3,2)

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(2 x+3 y=6\), point (0,5)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

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.