What Is The Slope Of The Line That Passes Through The Points (1, -6) And (-2, -8)?Write Your Answer In Simplest Form.
Introduction
In mathematics, the slope of a line is a fundamental concept that helps us understand the steepness or incline of a line. It is a measure of how much the line rises (or falls) vertically for every unit of horizontal distance it covers. The slope of a line can be calculated using the coordinates of two points on the line. In this article, we will explore how to find the slope of the line that passes through the points (1, -6) and (-2, -8).
What is Slope?
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. The slope is usually denoted by the letter 'm' and is calculated using the 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 the Line
To find the slope of the line that passes through the points (1, -6) and (-2, -8), we can use the formula above. We will substitute the coordinates of the two points into the formula and calculate the slope.
Step 1: Identify the Coordinates of the Two Points
The coordinates of the two points are (1, -6) and (-2, -8). We will use these coordinates to calculate the slope.
Step 2: Substitute the Coordinates into the Formula
We will substitute the coordinates of the two points into the formula:
m = (y2 - y1) / (x2 - x1) m = (-8 - (-6)) / (-2 - 1) m = (-8 + 6) / (-3) m = -2 / -3
Step 3: Simplify the Fraction
The fraction -2 / -3 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 1. However, we can simplify it further by changing the sign of the numerator and the denominator:
m = 2 / 3
Conclusion
The slope of the line that passes through the points (1, -6) and (-2, -8) is 2/3. This means that for every unit of horizontal distance the line covers, it rises 2 units vertically.
Real-World Applications of Slope
The concept of slope has many real-world applications. For example, in architecture, the slope of a roof is critical in determining the amount of rainfall that can be shed from the roof. In civil engineering, the slope of a road or a highway is important in determining the safety and efficiency of the road. In finance, the slope of a stock's price chart can be used to predict future price movements.
Tips for Calculating Slope
Calculating the slope of a line can be a straightforward process if you follow these tips:
- Make sure to identify the coordinates of the two points on the line.
- Substitute the coordinates into the formula for calculating slope.
- Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor.
- Change the sign of the numerator and the denominator if necessary.
Common Mistakes to Avoid
When calculating the slope of a line, there are several common mistakes to avoid:
- Make sure to use the correct coordinates of the two points on the line.
- Avoid substituting the coordinates into the formula incorrectly.
- Make sure to simplify the fraction correctly.
- Avoid changing the sign of the numerator and the denominator incorrectly.
Conclusion
In conclusion, the slope of the line that passes through the points (1, -6) and (-2, -8) is 2/3. This is a fundamental concept in mathematics that has many real-world applications. By following the tips and avoiding common mistakes, you can calculate the slope of a line with ease.
Frequently Asked Questions
Q: What is the slope of a horizontal line?
A: The slope of a horizontal line is 0.
Q: What is the slope of a vertical line?
A: The slope of a vertical line is undefined.
Q: How do I calculate the slope of a line using the coordinates of two points?
A: To calculate the slope of a line using the coordinates of two points, 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 greatest common divisor of two numbers?
A: The greatest common divisor of two numbers is the largest number that divides both numbers without leaving a remainder.
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.
Q: What is the difference between a positive and a negative slope?
A: A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right.
Introduction
In our previous article, we discussed the concept of slope and how to calculate it using the coordinates of two points on a line. However, we know that there are many more questions that you may have about slope. In this article, we will answer some of the most frequently asked questions about slope.
Q: What is the slope of a horizontal line?
A: The slope of a horizontal line is 0. This is because a horizontal line does not rise or fall, it simply extends horizontally.
Q: What is the slope of a vertical line?
A: The slope of a vertical line is undefined. This is because a vertical line does not have a rise or run, it simply extends up and down.
Q: How do I calculate the slope of a line using the coordinates of two points?
A: To calculate the slope of a line using the coordinates of two points, 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 greatest common divisor of two numbers?
A: The greatest common divisor of two numbers is the largest number that divides both numbers without leaving a remainder.
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.
Q: What is the difference between a positive and a negative slope?
A: A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls 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.
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.
Q: How do I determine the slope of a line from a graph?
A: To determine the slope of a line from a graph, you can use the following steps:
- Identify two points on the line.
- Calculate the rise (vertical change) and run (horizontal change) between the two points.
- Divide the rise by the run to get the slope.
Q: What is the slope of a line that passes through the origin?
A: The slope of a line that passes through the origin is the same as the slope of the line that passes through any other two points on the line.
Q: Can the slope of a line be a fraction?
A: Yes, the slope of a line can be a fraction. This occurs when the line is not horizontal or vertical.
Q: How do I calculate the slope of a line using the point-slope form?
A: To calculate the slope of a line using the point-slope form, you can use the following formula:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
Q: What is the slope of a line that passes through two points with the same x-coordinate?
A: The slope of a line that passes through two points with the same x-coordinate is undefined.
Q: Can the slope of a line be a decimal?
A: Yes, the slope of a line can be a decimal. This occurs when the line is not horizontal or vertical.
Q: How do I determine the slope of a line from a table of values?
A: To determine the slope of a line from a table of values, you can use the following steps:
- Identify two points on the line.
- Calculate the rise (vertical change) and run (horizontal change) between the two points.
- Divide the rise by the run to get the slope.
Q: What is the slope of a line that passes through two points with the same y-coordinate?
A: The slope of a line that passes through two points with the same y-coordinate is 0.
Q: Can the slope of a line be a negative fraction?
A: Yes, the slope of a line can be a negative fraction. This occurs when the line falls from left to right.
Q: How do I calculate the slope of a line using the slope-intercept form?
A: To calculate the slope of a line using the slope-intercept form, you can use the following formula:
y = mx + b
where m is the slope and b is the y-intercept.
Q: What is the slope of a line that passes through two points with the same x and y coordinates?
A: The slope of a line that passes through two points with the same x and y coordinates is undefined.
Q: Can the slope of a line be a positive fraction?
A: Yes, the slope of a line can be a positive fraction. This occurs when the line rises from left to right.
Q: How do I determine the slope of a line from a graph with a non-linear scale?
A: To determine the slope of a line from a graph with a non-linear scale, you can use the following steps:
- Identify two points on the line.
- Calculate the rise (vertical change) and run (horizontal change) between the two points.
- Divide the rise by the run to get the slope.
Q: What is the slope of a line that passes through two points with different x and y coordinates?
A: The slope of a line that passes through two points with different x and y coordinates is a fraction.
Q: Can the slope of a line be a negative decimal?
A: Yes, the slope of a line can be a negative decimal. This occurs when the line falls from left to right.
Q: How do I calculate the slope of a line using the equation of a circle?
A: To calculate the slope of a line using the equation of a circle, you can use the following formula:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Q: What is the slope of a line that passes through two points with the same x and y coordinates?
A: The slope of a line that passes through two points with the same x and y coordinates is undefined.
Q: Can the slope of a line be a positive decimal?
A: Yes, the slope of a line can be a positive decimal. This occurs when the line rises from left to right.
Q: How do I determine the slope of a line from a graph with a logarithmic scale?
A: To determine the slope of a line from a graph with a logarithmic scale, you can use the following steps:
- Identify two points on the line.
- Calculate the rise (vertical change) and run (horizontal change) between the two points.
- Divide the rise by the run to get the slope.
Q: What is the slope of a line that passes through two points with different x and y coordinates?
A: The slope of a line that passes through two points with different x and y coordinates is a fraction.
Q: Can the slope of a line be a negative fraction?
A: Yes, the slope of a line can be a negative fraction. This occurs when the line falls from left to right.
Q: How do I calculate the slope of a line using the equation of a parabola?
A: To calculate the slope of a line using the equation of a parabola, you can use the following formula:
y = ax^2 + bx + c
where a, b, and c are constants.
Q: What is the slope of a line that passes through two points with the same x and y coordinates?
A: The slope of a line that passes through two points with the same x and y coordinates is undefined.
Q: Can the slope of a line be a positive fraction?
A: Yes, the slope of a line can be a positive fraction. This occurs when the line rises from left to right.
Q: How do I determine the slope of a line from a graph with a polar coordinate system?
A: To determine the slope of a line from a graph with a polar coordinate system, you can use the following steps:
- Identify two points on the line.
- Calculate the rise (vertical change) and run (horizontal change) between the two points.
- Divide the rise by the run to get the slope.
Q: What is the slope of a line that passes through two points with different x and y coordinates?
A: The slope of a line that passes through two points with different x and y coordinates is a fraction.
Q: Can the slope of a line be a negative decimal?
A: Yes, the slope of a line can be a negative decimal. This occurs when the line falls from left to right.
Q: How do I calculate the slope of a line using the equation of an ellipse?
A: To calculate the slope of a line using the equation of an ellipse, you can use the following formula:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
where (h, k) is the center of the ellipse and a and b are the semi-major and semi-minor axes.
Q: What is the slope of a line that passes through two points with the same x and y coordinates?
A: The slope of a line that passes through two points with the same x and y coordinates is undefined.
Q: Can the slope of a line be a positive decimal?
A: Yes, the slope of a line can be a positive decimal. This occurs when the line rises from left to