Find The Slope Of The Line That Passes Through The Points (4, 2) And (2, 3). Simplify Your Answer And Write It As A Proper Fraction, Improper Fraction, Or Integer.
Introduction
In mathematics, the slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In this article, we will learn how to find the slope of a line that passes through two given points.
What is Slope?
The slope of a line is a fundamental concept in mathematics, and it is used to describe the steepness of a line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope is denoted by the letter 'm' and is calculated using the following formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points on the line.
Finding the Slope of a Line
To find the slope of a line that passes through two given points, we can use the formula above. Let's consider the points (4, 2) and (2, 3). We can plug these values into the formula to find the slope.
Step 1: Identify the Coordinates of the Two Points
The coordinates of the two points are (4, 2) and (2, 3). We can identify the x-coordinates as 4 and 2, and the y-coordinates as 2 and 3.
Step 2: Plug the Values into the Formula
Now that we have identified the coordinates of the two points, we can plug these values into the formula to find the slope.
m = (y2 - y1) / (x2 - x1) m = (3 - 2) / (2 - 4) m = 1 / -2
Step 3: Simplify the Answer
The answer we obtained is a fraction. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 1 and 2 is 1. Therefore, the simplified answer is:
m = -1/2
Step 4: Write the Answer as a Proper Fraction, Improper Fraction, or Integer
The answer we obtained is a proper fraction. However, we can also write it as an improper fraction or an integer. To write it as an improper fraction, we can multiply the numerator and the denominator by -1:
m = (-1/2) × (-1/-1) m = 1/2
To write it as an integer, we can multiply the numerator and the denominator by 2:
m = (1/2) × (2/2) m = 1
Conclusion
In this article, we learned how to find the slope of a line that passes through two given points. We used the formula m = (y2 - y1) / (x2 - x1) to find the slope of the line that passes through the points (4, 2) and (2, 3). We simplified the answer and wrote it as a proper fraction, improper fraction, or integer.
Real-World Applications of Slope
The concept of slope has many real-world applications. For example, in architecture, the slope of a roof is an important consideration when designing a building. In engineering, the slope of a road or a bridge is critical in ensuring the safety of vehicles and pedestrians. In finance, the slope of a stock's price chart can indicate the direction of the market.
Common Mistakes to Avoid
When finding the slope of a line, there are several common mistakes to avoid. One mistake is to confuse the x and y coordinates. Another mistake is to forget to simplify the answer. Finally, a common mistake is to write the answer as a decimal instead of a fraction.
Tips and Tricks
When finding the slope of a line, there are several tips and tricks to keep in mind. One tip is to use the formula m = (y2 - y1) / (x2 - x1) to find the slope. Another tip is to simplify the answer by dividing both the numerator and the denominator by their greatest common divisor (GCD). Finally, a tip is to write the answer as a proper fraction, improper fraction, or integer.
Conclusion
Q: What is the slope of a line?
A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Q: How do I find the slope of a line?
A: To find the slope of a line, you can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points on the line.
Q: What is the difference between a positive and negative slope?
A: A positive slope indicates that the line is rising from left to right, while a negative slope indicates that the line is falling from left to right.
Q: Can the slope of a line be zero?
A: Yes, the slope of a line can be zero. This occurs when the line is horizontal, meaning that it does not rise or fall.
Q: Can the slope of a line be undefined?
A: Yes, the slope of a line can be undefined. This occurs when the line is vertical, meaning that it does not have a horizontal change.
Q: How do I simplify a fraction?
A: To simplify a fraction, you can divide both the numerator and the denominator by their greatest common divisor (GCD).
Q: Can I write the slope of a line as a decimal?
A: Yes, you can write the slope of a line as a decimal. However, it is often more convenient to write it as a fraction.
Q: What is the slope of a line that passes through the points (0, 0) and (3, 4)?
A: To find the slope of this line, we can use the formula m = (y2 - y1) / (x2 - x1). Plugging in the values, we get:
m = (4 - 0) / (3 - 0) m = 4/3
Q: What is the slope of a line that passes through the points (2, 5) and (4, 3)?
A: To find the slope of this line, we can use the formula m = (y2 - y1) / (x2 - x1). Plugging in the values, we get:
m = (3 - 5) / (4 - 2) m = -2/2 m = -1
Q: Can I find the slope of a line using a calculator?
A: Yes, you can find the slope of a line using a calculator. Simply enter the coordinates of the two points and use the calculator's slope function.
Q: What is the slope of a line that passes through the points (1, 2) and (3, 4)?
A: To find the slope of this line, we can use the formula m = (y2 - y1) / (x2 - x1). Plugging in the values, we get:
m = (4 - 2) / (3 - 1) m = 2/2 m = 1
Conclusion
In conclusion, the slope of a line is an important concept in mathematics. It is used to describe the steepness of a line and has many real-world applications. By following the steps outlined in this article, you can find the slope of a line that passes through two given points. Remember to simplify the answer and write it as a proper fraction, improper fraction, or integer.