/*! 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 Find the slope of the line satis... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 40

Find the slope of the line satisfying the given conditions. through \((-2,4)\) and \((6,4)\)

Short Answer

Expert verified
The slope of the line is 0.

Step by step solution

01

Identify the coordinates

The points given are (-2, 4) and (6, 4). Label them as (x_1, y_1) and (x_2, y_2) respectively.
02

Recall the slope formula

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using this formula, substitute the given coordinates: (x_1, y_1) = (-2, 4) and (x_2, y_2) = (6, 4).
03

Substitute the values

After substitution, the formula becomes \(m = \frac{4 - 4}{6 + 2}\).
04

Simplify the expression

Simplify the numerator and denominator: \(m = \frac{0}{8}\).
05

Interpret the result

Since \(\frac{0}{8} = 0\), the slope of the line is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic techniques to study geometric properties. It involves plotting points, lines, and curves on a coordinate plane. The plane is a two-dimensional surface defined by an x-axis (horizontal) and a y-axis (vertical). Each point on this plane is represented with coordinates \((x, y)\), where \(x\) tells us the horizontal position and \(y\) tells us the vertical position. In this exercise, the points given are \((-2, 4)\) and \(6, 4)\). These points are plotted based on their respective coordinates and lie on the same horizontal line.
Slope Formula
The slope of a line, denoted by \(m\), is a measure of its steepness and direction. To find the slope of a line passing through two points, we use the slope formula:
\[\text{slope} = m = \frac{y_2 - y_1}{x_2 - x_1} \]]. Here:
  • \(x_1\) and \(y_1\) are the coordinates of the first point.
  • \(x_2\) and \(y_2\) are the coordinates of the second point.
The slope formula calculates the ratio of the vertical change (difference in y-values) to the horizontal change (difference in x-values). In the given exercise, both points are \( (-2, 4)\) and \( (6, 4)\). Using these coordinates :
\[m = \frac{4 - 4}{6 + 2} \].
Linear Equations
Linear equations represent straight lines on a coordinate plane. These equations are typically written in the form \y = mx + b\, where:
  • \(m\) is the slope of the line
  • \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
In the exercise given, the slope \ (m = 0) \ was found, which means the line is horizontal. For any horizontal line, the y-value remains constant, so the equation becomes \ y = b \ (where \ b \ is the y-value of the coordinates). Since both points given in the problem have a y-coordinate of 4, the linear equation for the line is \ y = 4 \.

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Most popular questions from this chapter

For each of the functions in Exercises \(33-46,\) find ( \(a\) ) \(f(x+h),\) (b) \(f(x+h)-f(x)\) and \((c) \frac{f(x+h)-f(x)}{h} .\) See Example 4. $$f(x)=1-x$$

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