Which Expression Can Be Used To Determine The Slope Of The Line That Passes Through The Points \[$(-7, 3)\$\] And \[$ (1, -9) \$\]?A. \[$\frac{1-(-7)}{-9-3}\$\]B. \[$\frac{1+(-7)}{-9+3}\$\]C.
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 crucial concept in geometry, algebra, and calculus. In this article, we will explore how to determine the slope of a line that passes through two given points. We will use the concept of the slope formula and apply it to two different scenarios.
The Slope Formula
The slope formula is a mathematical expression that helps us calculate the slope of a line that passes through two points. The formula is given by:
m = (y2 - y1) / (x2 - x1)
where m is the slope of the line, and (x1, y1) and (x2, y2) are the coordinates of the two points.
Scenario 1: Using the Correct Formula
Let's consider the points (-7, 3) and (1, -9). We want to determine the slope of the line that passes through these two points. To do this, we will use the slope formula.
The correct formula is:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (-7, 3) and (x2, y2) = (1, -9).
Plugging in the values, we get:
m = (-9 - 3) / (1 - (-7)) m = (-12) / (8) m = -1.5
Therefore, the slope of the line that passes through the points (-7, 3) and (1, -9) is -1.5.
Scenario 2: Using the Incorrect Formula
Now, let's consider the points (-7, 3) and (1, -9) again. However, this time, we will use the incorrect formula:
m = (x2 - x1) / (y2 - y1)
where (x1, y1) = (-7, 3) and (x2, y2) = (1, -9).
Plugging in the values, we get:
m = (1 - (-7)) / (-9 - 3) m = (8) / (-12) m = -0.67
As you can see, the result is different from the correct result. This is because the formula is incorrect.
Conclusion
In conclusion, the slope of a line that passes through two points can be determined using the slope formula. The correct formula is:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points. We should always use the correct formula to avoid errors.
Common Mistakes to Avoid
When determining the slope of a line, there are several common mistakes to avoid:
- Using the incorrect formula
- Swapping the x and y coordinates
- Not plugging in the correct values
- Not simplifying the expression
By avoiding these common mistakes, we can ensure that we get the correct result.
Real-World Applications
The concept of slope is used in many real-world applications, such as:
- Building design: Architects use the concept of slope to design buildings that are stable and safe.
- Road construction: Engineers use the concept of slope to design roads that are safe and efficient.
- Surveying: Surveyors use the concept of slope to determine the elevation of a point on the Earth's surface.
Q: What is the slope of a line?
A: The slope of a line is a measure of how steep the line 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 calculate the slope of a line?
A: To calculate the slope of a line, you can use the slope formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Q: What is the difference between the slope and the y-intercept?
A: The slope is a measure of how steep the line is, while the y-intercept is the point where the line intersects the y-axis. The slope is calculated as the ratio of the vertical change to the horizontal change, while the y-intercept is the value of y when x is equal to 0.
Q: Can I use the slope formula to find the equation of a line?
A: Yes, you can use the slope formula to find the equation of a line. Once you have the slope and one point on the line, you can use the point-slope form of a linear equation to find the equation of the line.
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
Q: Can I use the slope formula to find the equation of a line if I have two points?
A: Yes, you can use the slope formula to find the equation of a line if you have two points. Once you have the slope, you can use the two points to find the y-intercept and then use the slope-intercept form of a linear equation to find the equation of the line.
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is:
y = mx + b
where m is the slope and b is the y-intercept.
Q: Can I use the slope formula to find the equation of a line if I have a graph?
A: Yes, you can use the slope formula to find the equation of a line if you have a graph. Once you have the slope, you can use the graph to find the y-intercept and then use the slope-intercept form of a linear equation to find the equation of the line.
Q: What are some common mistakes to avoid when using the slope formula?
A: Some common mistakes to avoid when using the slope formula include:
- Swapping the x and y coordinates
- Not plugging in the correct values
- Not simplifying the expression
- Using the incorrect formula
Q: Can I use the slope formula to find the equation of a line if I have a table of values?
A: Yes, you can use the slope formula to find the equation of a line if you have a table of values. Once you have the slope, you can use the table of values to find the y-intercept and then use the slope-intercept form of a linear equation to find the equation of the line.
Q: What are some real-world applications of the slope formula?
A: Some real-world applications of the slope formula include:
- Building design: Architects use the slope formula to design buildings that are stable and safe.
- Road construction: Engineers use the slope formula to design roads that are safe and efficient.
- Surveying: Surveyors use the slope formula to determine the elevation of a point on the Earth's surface.
In conclusion, the slope formula is a powerful tool for determining the slope of a line. By understanding how to use the slope formula, you can apply it to a variety of real-world situations and solve problems in mathematics and engineering.