Give The Slope Of Y + 2 = 3 ( X − 7 Y + 2 = 3(x - 7 Y + 2 = 3 ( X − 7 ] And A Point On The Line.A. The Slope Is 3 And (7, -2) Is On The Line.B. The Slope Is 2 And (7, 3) Is On The Line.C. The Slope Is 3 And (-7, 2) Is On The Line.D. The Slope Is 7 And (3, 2) Is On The Line.
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. In this article, we will explore the concept of slope and how to find it in a given equation of a line.
What is the Slope of a Line?
The slope of a line is denoted by the letter 'm' and is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. It can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Finding the Slope of a Line from an Equation
To find the slope of a line from its equation, we need to rewrite the equation in the slope-intercept form, which is:
y = mx + b
where m is the slope and b is the y-intercept.
Let's consider the given equation:
y + 2 = 3(x - 7)
To rewrite this equation in the slope-intercept form, we need to isolate y on one side of the equation.
Rewriting the Equation
y + 2 = 3(x - 7)
y + 2 = 3x - 21
y = 3x - 23
Now that we have the equation in the slope-intercept form, we can see that the slope (m) is 3.
Finding a Point on the Line
To find a point on the line, we can substitute a value of x into the equation and solve for y. Let's substitute x = 0 into the equation:
y = 3(0) - 23
y = -23
So, one point on the line is (0, -23).
However, we are given a different point (7, -2) as an option. To verify if this point is on the line, we can substitute x = 7 and y = -2 into the equation:
y + 2 = 3(x - 7)
-2 + 2 = 3(7 - 7)
0 = 0
This equation is true, which means that the point (7, -2) is indeed on the line.
Conclusion
In conclusion, the slope of the line y + 2 = 3(x - 7) is 3, and a point on the line is (7, -2).
Answer
The correct answer is:
A. The slope is 3 and (7, -2) is on the line.
Additional Examples
Here are a few more examples of finding the slope of a line from its equation:
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y = 2x + 1
The slope is 2, and a point on the line is (0, 1).
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y = -x + 3
The slope is -1, and a point on the line is (0, 3).
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y = 4x - 2
The slope is 4, and a point on the line is (0, -2).
Tips and Tricks
Here are a few tips and tricks to help you find the slope of a line from its equation:
- Make sure to rewrite the equation in the slope-intercept form (y = mx + b) before finding the slope.
- Use the formula m = (y2 - y1) / (x2 - x1) to find the slope if you are given two points on the line.
- Check if the point you are given is on the line by substituting the values of x and y into the equation.
In this article, we will answer some frequently asked questions about the slope of a line.
Q: What is the slope of a line?
A: The slope of a line is a measure of how much the line rises (or falls) vertically for every unit of horizontal distance it covers. 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 from its equation?
A: To find the slope of a line from its equation, you need to rewrite the equation in the slope-intercept form (y = mx + b) and then identify the value of m, which is the slope.
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I find the slope of a line if I am given two points on the line?
A: To find the slope of a line if you are given two points on the line, you can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the two points on the line.
Q: What is the difference between the slope and the y-intercept?
A: The slope (m) is a measure of how much the line rises (or falls) vertically for every unit of horizontal distance it covers, while the y-intercept (b) is the point where the line intersects the y-axis.
Q: Can the slope of a line be negative?
A: Yes, the slope of a line can be negative. A negative slope indicates that the line falls as it moves from left to right.
Q: Can the slope of a line be zero?
A: Yes, the slope of a line can be zero. A slope of zero indicates that the line is horizontal and does not rise or fall as it moves from left to right.
Q: Can the slope of a line be undefined?
A: Yes, the slope of a line can be undefined. An undefined slope indicates that the line is vertical and does not have a slope in the classical sense.
Q: How do I determine if a line is parallel or perpendicular to another line?
A: To determine if a line is parallel or perpendicular to another line, you can compare their slopes. If the slopes are equal, the lines are parallel. If the product of the slopes is -1, the lines are perpendicular.
Q: Can a line have multiple slopes?
A: No, a line can only have one slope. The slope of a line is a unique value that describes the line's steepness or incline.
Q: Can a line have a slope of infinity?
A: No, a line cannot have a slope of infinity. A line with a slope of infinity would be a vertical line, but the slope of a vertical line is undefined, not infinity.
Conclusion
In conclusion, the slope of a line is a fundamental concept in mathematics that helps us understand the steepness or incline of a line. By following the tips and tricks outlined in this article, you can easily find the slope of a line from its equation and answer frequently asked questions about the slope of a line.
Additional Resources
Here are some additional resources that you may find helpful:
- Khan Academy: Slope of a Line
- Mathway: Slope of a Line
- Wolfram Alpha: Slope of a Line
By following these resources and practicing the concepts outlined in this article, you can become proficient in finding the slope of a line and answering frequently asked questions about the slope of a line.