/*! 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 40 Contains the point \((-4,7)\) an... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 40

Contains the point \((-4,7)\) and is perpendicular to the \(x\) axis

Short Answer

Expert verified
The equation of the line is x = -4.

Step by step solution

01

Identify the Key Condition

The line needs to be perpendicular to the x-axis. A line perpendicular to the x-axis is a vertical line.
02

Understand the Characteristics of Vertical Lines

Vertical lines have a constant x-coordinate, which means that the x-value is the same for all points on the line.
03

Determine the Equation of the Line

Since the line is vertical and contains the point (-4,7), it must have the x-coordinate of -4 for all its points. Therefore, the equation of the line is x = -4.

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.

Vertical Lines
Vertical lines are a special type of line in coordinate geometry. They have a unique characteristic where all points along the line share the same x-coordinate. This means the line does not have a slope like diagonal lines do. Since there is no change in the x-values, vertical lines run parallel to the y-axis. In simpler terms, imagine drawing a straight line that goes up and down without tilting to the sides.
Here are some key points about vertical lines:
  • A vertical line has an undefined slope. This is because slope is calculated as the rise over the run (change in y over change in x), and in a vertical line, the change in x is zero, which makes the slope undefined.
  • The equation of a vertical line can always be represented as \(x = a\), where "a" is the constant x-coordinate for every point on the line.
  • Vertical lines are always perpendicular to horizontal lines, which run parallel to the x-axis.
Equation of a Line
An equation of a line is a mathematical way to represent all the points along a line on a coordinate plane. It provides a rule that every point on the line follows. The most common form is the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
However, vertical lines do not fit into this form because their slopes are undefined. For a vertical line, the equation is different:
  • For vertical lines, the equation is simply \(x = a\), where "a" is the constant value of the x-coordinate for the line. In this case, if a line passes through the point \((-4, 7)\), the equation is \(x = -4\).
This means that no matter where you move along this line, the x-coordinate remains the same, reinforcing the concept of verticality. It doesn't intersect the y-axis, and its definition is easy and straightforward.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, merges algebra with geometry using a coordinate system. It allows us to express geometric properties through algebraic equations and vice versa.
Coordinate geometry involves:
  • Points, which are represented by pairs of numbers (x, y), showing their position on the Cartesian plane.
  • Lines, which are defined by equations showing the relation between x and y coordinates of any point lying on the line.
  • Shapes, which can be graphically represented and analyzed using equations, often revealing deeper properties like sizes and angles.
A key strength of coordinate geometry is its ability to help us understand relationships between geometric figures and their equations. For instance, it clarifies when two lines are parallel, perpendicular, or intersecting. The simplicity and power of coordinate geometry allow us to solve problems involving distances, midpoints, and slopes, and understand the nature of vertical and horizontal lines effortlessly.

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

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

A company uses 7 pounds of fertilizer for a lawn that measures 5000 square feet and 12 pounds for a lawn that measures 10,000 square feet. Let \(y\) represent the pounds of fertilizer and \(x\) the square footage of the lawn.

The grade of a highway up a hill is \(30 \%\). How much change in horizontal distance is there if the vertical height of the hill is 75 feet?

Use the DRAW feature of your graphing calculator to draw each of the following. (a) A line segment between \((-2,-4)\) and \((-2,5)\) (b) A line segment between \((2,2)\) and \((5,2)\) (c) A line segment between \((2,3)\) and \((5,7)\) (d) A triangle with vertices at \((1,-2),(3,4)\), and \((-3,6)\)

\(x\) intercept of \(-3\) and slope of \(-\frac{5}{8}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.