Problem 6
Question
Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. $$(4,-1)\( and \)(3,-1)$$
Step-by-Step Solution
Verified Answer
The slope of the line passing through the points (4, -1) and (3, -1) is 0. This implies that the line is horizontal.
1Step 1: Identify the points
The points given are (4, -1) and (3, -1). Let's denote them as (x1, y1) and (x2, y2) respectively.
2Step 2: Apply the slope formula
The formula for the slope of a line between two points is given by \(m = \frac{y2 - y1}{x2 - x1}\). Using the given points, the slope becomes \(m = \frac{-1 - (-1)}{3 - 4}\).
3Step 3: Simplify the obtained expression
Simplifying the obtained expression, we get \(m = \frac{0}{-1} = 0\).
4Step 4: Interpret the slope
The slope 0 means that the line is horizontal.
Key Concepts
Coordinate GeometryAlgebraic FormulasHorizontal Line
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, bridges algebra and geometry by using a coordinate system. This system facilitates the process of finding properties of figures such as distance, midpoint, gradients, and area through an algebraic approach.
In the context of the slope of a line, coordinate geometry allows us to calculate how steep a line is when it crosses two given points on the Cartesian plane. Points on this plane are described by ordered pairs \( (x, y) \), where 'x' represents the horizontal distance from the origin and 'y' the vertical. The slope, or gradient, is a measure of how fast y increases as x increases, indicating the direction of the line.
In the context of the slope of a line, coordinate geometry allows us to calculate how steep a line is when it crosses two given points on the Cartesian plane. Points on this plane are described by ordered pairs \( (x, y) \), where 'x' represents the horizontal distance from the origin and 'y' the vertical. The slope, or gradient, is a measure of how fast y increases as x increases, indicating the direction of the line.
Algebraic Formulas
Algebraic formulas comprise a variety of equations that solve numerous mathematical problems through the manipulation of symbols and numbers. One essential formula used in coordinate geometry is the slope formula:
\[ m = \frac{y2 - y1}{x2 - x1} \] where \((x1, y1)\) and \((x2, y2)\) represent two distinct points on a line.
To determine the slope \(m\), you subtract the y-coordinate of the first point from the y-coordinate of the second point and divide the result by the difference between their x-coordinates. \(m\) indicates the line's steepness; a greater absolute value means a steeper slope. A positive slope implies that the line rises as it moves to the right, while a negative slope signifies that the line falls.
\[ m = \frac{y2 - y1}{x2 - x1} \] where \((x1, y1)\) and \((x2, y2)\) represent two distinct points on a line.
To determine the slope \(m\), you subtract the y-coordinate of the first point from the y-coordinate of the second point and divide the result by the difference between their x-coordinates. \(m\) indicates the line's steepness; a greater absolute value means a steeper slope. A positive slope implies that the line rises as it moves to the right, while a negative slope signifies that the line falls.
Horizontal Line
The special nature of a horizontal line in coordinate geometry is that it has a slope of 0. This is because the y-coordinates of any two points on a horizontal line are the same, which means that their difference is zero.
As a result, the formula for slope given by \[ m = \frac{y2 - y1}{x2 - x1} \] simplifies to \( m = \frac{0}{x2 - x1} = 0 \), regardless of the difference between the x-coordinates.
A horizontal line suggests a constant y-value for all x-values, which is the hallmark of 'no change' in elevation or levelness. Such lines are often used to represent the concept of stability or equilibrium in diagrams, graphs, and real-world applications.
As a result, the formula for slope given by \[ m = \frac{y2 - y1}{x2 - x1} \] simplifies to \( m = \frac{0}{x2 - x1} = 0 \), regardless of the difference between the x-coordinates.
A horizontal line suggests a constant y-value for all x-values, which is the hallmark of 'no change' in elevation or levelness. Such lines are often used to represent the concept of stability or equilibrium in diagrams, graphs, and real-world applications.
Other exercises in this chapter
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