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Magazine Chapter 5: Problem 11
Find the slope of the line that passes through the given points. $$(0.4,0.5),(-0.2,0.2)$$
Short Answer
Expert verified
The slope of the line is 0.5.
Step by step solution
01
Understand the Slope Formula
The formula to find 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 calculates how much the \(y\)-value changes for a given change in \(x\)-value, effectively describing the steepness of the line.
02
Identify Coordinates
Identify and label the given points. The first point \((0.4, 0.5)\) is \((x_1, y_1)\) and the second point \((-0.2, 0.2)\) is \((x_2, y_2)\). This gives us: \(x_1 = 0.4\), \(y_1 = 0.5\), \(x_2 = -0.2\), \(y_2 = 0.2\).
03
Substitute into the Slope Formula
Substitute the identified coordinates into the slope formula: \[ m = \frac{0.2 - 0.5}{-0.2 - 0.4} \].
04
Calculate Numerator and Denominator Separately
First, calculate the numerator: \(0.2 - 0.5 = -0.3\).Then, calculate the denominator: \(-0.2 - 0.4 = -0.6\).
05
Divide Numerator by Denominator
Now divide the numerator by the denominator to find the slope: \[ m = \frac{-0.3}{-0.6} = 0.5 \].
06
Interpret the Result
The calculated slope \(m = 0.5\) means that for every unit increase in \(x\), \(y\) increases by 0.5 units. The line is moderately sloping upward.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope Formula
When determining the slope of a line from two points, the slope formula is your key tool. This helps you describe the line鈥檚 steepness or direction. The slope (\(m\)) of a line that cuts through two distinct points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1} \]
- The numerator, \(y_2 - y_1\), reflects the vertical change known as the "rise," which shows how much the \(y\)-coordinate changes.
- The denominator, \(x_2 - x_1\), reflects the horizontal change, or the "run," showing how much the \(x\)-coordinate changes.
Coordinate Geometry
Coordinate geometry, or "analytic geometry," brings the algebraic and geometric worlds together, allowing us to study geometry using a coordinate system. This concept involves representing lines, points, and shapes using coordinates.Here's how it relates to the slope:
- Coordinates help us define specific points on a plane. In this system, a point is described using a pair of values, \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position.
- Using these coordinates, we can assess the relationship between different points, like finding the slope.
Calculating Slope from Points
Calculating the slope from points involves a few simple steps, making it accessible with basic arithmetic. Let's break down the process:First, identify the coordinates of your two points, labeling them as \((x_1, y_1)\) and \((x_2, y_2)\). With these values:
- Use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
- Subtract the \(y\)-coordinates: \(y_2 - y_1\), to find the "rise." This gives the vertical change.
- Subtract the \(x\)-coordinates: \(x_2 - x_1\), to find the "run," or the horizontal change.
- Divide the rise by the run to find the slope \(m\).
