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Chapter 5: Problem 31

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((9.62,8.77)\) and \((-1.4,8.77)\)

Short Answer

Expert verified
The slope is 0.

Step by step solution

01

Recall the slope formula

The slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
02

Substitute the given points into the formula

Given points are \( (9.62, 8.77) \) and \( (-1.4, 8.77) \). Substituting these points into the formula: \[ m = \frac{8.77 - 8.77}{-1.4 - 9.62} \]
03

Simplify the numerator

Calculate the numerator: \[ 8.77 - 8.77 = 0 \]
04

Simplify the denominator

Calculate the denominator: \[ -1.4 - 9.62 = -11.02 \]
05

Compute the slope

Perform the division: \[ m = \frac{0}{-11.02} = 0 \]

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Key Concepts

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

Slope Formula
The slope of a line indicates its steepness and is essential in understanding linear equations and algebraic functions. The slope formula is given by: \( m = \frac{y_2 - y_1}{x_2 - x_1} \) where \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of any two points on the line. This formula calculates the ratio of the vertical change (difference in y-coordinates) to the horizontal change (difference in x-coordinates) between two points. \
\
This calculation helps determine whether a line is rising, falling, or remains constant as it moves from left to right \
Coordinates
Coordinates are pairs of numbers that define a point's location on a plane, typically represented as (x, y). The first number, x, denotes horizontal position while the second number, y, specifies the vertical position. \
\
In the given problem, the points \((9.62, 8.77)\) and \((-1.4, 8.77)\) are used. Both of these points share the same y-coordinate, 8.77, indicating that the line passes through these points horizontally because there is no vertical change. \
\
Understanding these points in the coordinate system is fundamental to determining how a line behaves.
Linear Equations
Linear equations form the foundation of many algebraic concepts. An equation is called linear if it forms a straight line when plotted on a graph. The standard format for linear equations is \(y = mx + c\), where: \
    \
  • y represents the dependent variable \
  • x is the independent variable \
  • m is the slope \
  • c is the y-intercept, the point where the line crosses the y-axis\
\
Identifying the slope from two points helps in building and understanding linear equations. For our particular problem, the fact that the slope \((m)\) is zero indicates a horizontal line, emphasizing no vertical change as x varies.
Algebraic Functions
Algebraic functions involve operations on variables, and in this context, we focus on linear functions, a subset of algebraic functions. These functions are crucial as they describe relationships with constant rates of change. \
\
The slope formula, \( m = \frac{y_2 - y_1}{x_2 - x_1} \), embodies these concepts by measuring this constant rate, or the gradient of the line. \
\
In our problem, solving the slope provides insight into the relationship defined by the linear function passing through \((9.62, 8.77)\) and \((-1.4, 8.77)\). This particular relationship is constant since there is no vertical shift across these points, asserting the zero slope.

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Most popular questions from this chapter

How could you use the idea of slope to show that the three points \((-1,-2)\) \((2,0),\) and \((5,2)\) all lie on a straight line?

Plot a few points that satisfy the equation \(y=x^{3} .\) Do you think the graph of this equation is a straight line? Explain.

A slanted roof on a house rises 2.8 meters over a horizontal distance of 6.2 meters. Find the grade (or pitch) of the roof.

Write an equation of the line satisfying the given conditions. Vertical line passing through \((2,0)\)

Given the equation \(3 x+2 y=12,\) complete the given ordered pairs: $$(2, \quad) (0, \quad) \quad(\quad,-3) \quad(\quad, 0)$$
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