Problem 65
Question
For each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. (2,4) and (1,4)
Step-by-Step Solution
Verified Answer
The slope is 0, and the line is horizontal.
1Step 1: Identify the formula for the slope
The formula for finding the slope (m) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. This formula will be used to calculate the slope between the given points \((2, 4)\) and \((1, 4)\).
2Step 2: Substitute the coordinates into the formula
Substitute \((x_1, y_1) = (2, 4)\) and \((x_2, y_2) = (1, 4)\) into the slope formula: \[ m = \frac{4 - 4}{1 - 2} \].
3Step 3: Calculate the slope
Simplify the expression: \[ m = \frac{0}{-1} = 0 \]. The slope of the line passing through the points \((2, 4)\) and \((1, 4)\) is 0.
4Step 4: Determine the direction of the line
Since the slope \(m = 0\), the line is horizontal. A slope of zero indicates the y-values do not change as the x-values change, resulting in a horizontal line.
Key Concepts
Horizontal LineSlope Calculation FormulaCoordinate Geometry
Horizontal Line
A horizontal line is a straight line that runs from left to right across the coordinate plane. This line has the unique characteristic that all points on it have the same y-coordinate. As such, the y-values do not change no matter how much you move along the x-axis. This results in a slope of 0.
For example, consider the points (2, 4) and (1, 4). Since the y-coordinates are both 4, any line drawn through these points will lie perfectly flat, parallel to the x-axis. A line like this does not rise or fall as you move along it; it remains level.
Horizontal lines are often seen in real-world scenarios, like the horizon or calm bodies of water. In mathematical terms, they are represented by equations such as y = c, where c is a constant.
For example, consider the points (2, 4) and (1, 4). Since the y-coordinates are both 4, any line drawn through these points will lie perfectly flat, parallel to the x-axis. A line like this does not rise or fall as you move along it; it remains level.
Horizontal lines are often seen in real-world scenarios, like the horizon or calm bodies of water. In mathematical terms, they are represented by equations such as y = c, where c is a constant.
Slope Calculation Formula
The slope of a line measures its steepness and direction. The slope calculation formula is a critical part of coordinate geometry. It helps us determine whether a line is increasing, decreasing, horizontal, or vertical.
The formula to find the slope (m) between two points \(x_1, y_1\) and \(x_2, y_2\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula calculates the change in the y-coordinates divided by the change in the x-coordinates.
A positive slope means the line slopes upwards from left to right, while a negative slope indicates it slopes downwards. When the result is 0, the line is horizontal, and if it's undefined, the line is vertical.
For example, using points (2, 4) and (1, 4), if we substitute into the formula, we find the slope is 0. This means there is no vertical change as the x-values change, confirming it is a horizontal line.
The formula to find the slope (m) between two points \(x_1, y_1\) and \(x_2, y_2\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula calculates the change in the y-coordinates divided by the change in the x-coordinates.
A positive slope means the line slopes upwards from left to right, while a negative slope indicates it slopes downwards. When the result is 0, the line is horizontal, and if it's undefined, the line is vertical.
For example, using points (2, 4) and (1, 4), if we substitute into the formula, we find the slope is 0. This means there is no vertical change as the x-values change, confirming it is a horizontal line.
Coordinate Geometry
Coordinate geometry is a branch of geometry where we analyze and prove the properties of geometric figures using coordinates. It involves placing geometric figures in the plane using a coordinate system, typically a Cartesian plane.
On this plane, positions are determined by pairs of numbers \(x, y\) known as coordinates, which tell you how far to move on the x-axis (horizontal) and y-axis (vertical). This makes it easier to perform calculations, such as finding the slope of a line passing through two points.
Knowing how to utilize coordinate geometry allows you to explore geometric concepts algebraically, like the properties of slopes and how points in a line relate to each other.
Problems involving finding the slope, like the one with points (2, 4) and (1, 4), heavily rely on this method by plotting points on the plane and using formulas to derive more information about their relationships. Understanding coordinate geometry provides a solid foundation for more advanced mathematical concepts.
On this plane, positions are determined by pairs of numbers \(x, y\) known as coordinates, which tell you how far to move on the x-axis (horizontal) and y-axis (vertical). This makes it easier to perform calculations, such as finding the slope of a line passing through two points.
Knowing how to utilize coordinate geometry allows you to explore geometric concepts algebraically, like the properties of slopes and how points in a line relate to each other.
Problems involving finding the slope, like the one with points (2, 4) and (1, 4), heavily rely on this method by plotting points on the plane and using formulas to derive more information about their relationships. Understanding coordinate geometry provides a solid foundation for more advanced mathematical concepts.
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