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Magazine Chapter 8: Problem 16
Find the slope of the line determined by each pair of points. $$(-4,-5),(-4,9)$$
Short Answer
Expert verified
The slope is undefined (vertical line).
Step by step solution
01
Identify Coordinates
First, identify the coordinates of the given points. The points are \((-4, -5)\) and \((-4, 9)\). Here, Point 1 is \((-4, -5)\) and Point 2 is \((-4, 9)\).
02
Apply the Slope Formula
The formula to find the slope \(m\) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
03
Substitute Coordinates into Slope Formula
Substitute the coordinates of the given points into the slope formula: For Point 1 \((x_1, y_1) = (-4, -5)\) and for Point 2 \((x_2, y_2) = (-4, 9)\), substitute into the formula: \[ m = \frac{9 - (-5)}{-4 - (-4)} \].
04
Perform the Calculations
First, simplify the numerator: \(9 - (-5) = 9 + 5 = 14\). Then the denominator becomes: \(-4 - (-4) = -4 + 4 = 0 \). So, the slope is \( \frac{14}{0} \).
05
Determine the Nature of Slope
Since the denominator of the slope calculation is 0, the slope is undefined. This means the line is vertical.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vertical Line
A vertical line is a straight line that runs up and down, parallel to the y-axis on a coordinate plane. All the points on a vertical line have the same x-coordinate, which means the line does not slant left or right, just straight through the top and bottom of the plane.
- Example: If you have two points, \((-4, -5)\) and \((-4, 9)\), they are both on the vertical line \(x = -4\) since their x-coordinates are identical.
- Visual Aspect: Imagine drawing a line through these points - it would look like a straight line that never touches the horizontal (x-axis) except at one fixed point.
Undefined Slope
The undefined slope is a concept you encounter with vertical lines. Slope generally refers to the steepness or tilt of a line and is calculated by the change in y over the change in x between two points.
- Formula: The slope \(m\) is described as \( m = \frac{y_2 - y_1}{x_2 - x_1}\).
- For a vertical line, the change in x, or \(x_2 - x_1\), is zero.
Coordinates of Points
When dealing with the slope of a line, the coordinates of points serve as a crucial starting point. They tell us exactly where each point lies on the graph.
- Definition: Each point on a two-dimensional plane is denoted by \((x, y)\), where \(x\) is the horizontal position, and \(y\) is the vertical position.
- Purpose: Knowing the coordinates allows us to find the slope or to understand the line's direction and length.
- Example: In our original problem, we have points \((-4, -5)\) and \((-4, 9)\). Their x-coordinates are identical (\(-4\)), indicating the line connecting them is indeed vertical.
Slope Formula
The slope formula is essential for determining how a line slants on a graph. It essentially quantifies how a line moves up or down as it moves from left to right across the graph.
- Formula: The formula for the slope \(m\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
- Process: Plug the respective \(x\) and \(y\) coordinates into the formula to find the slope.
- Application: For the points \((-4, -5)\) and \((-4, 9)\), you substitute directly into the slope formula, yielding \( m = \frac{9 - (-5)}{-4 - (-4)} = \frac{14}{0} \).
