Find The Point-slope Equation Of The Line Through The Points \[$(4,1)\$\] And \[$(2,-3)\$\].A. \[$y - 3 = 2(x - 2)\$\]B. \[$y - 1 = 2(x - 4)\$\]C. \[$y - 3 = -2(x + 2)\$\]D. \[$y - 1 = -2(x - 4)\$\]
Introduction
In mathematics, the point-slope equation of a line is a way to express the equation of a line using the coordinates of two points on the line. This equation is useful for finding the equation of a line when you know two points that the line passes through. In this article, we will learn how to find the point-slope equation of a line through two given points.
What is the Point-Slope Equation?
The point-slope equation of a line is given by the formula:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, m is the slope of the line, and (x, y) is any point on the line.
Finding the Slope of the Line
To find the point-slope equation of a line, we need to find the slope of the line first. The slope of a line is a measure of how steep the line is. It can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Example: Finding the Slope of the Line
Let's say we have two points (4, 1) and (2, -3) on the line. We can use the formula above to find the slope of the line.
m = (-3 - 1) / (2 - 4) m = -4 / -2 m = 2
So, the slope of the line is 2.
Finding the Point-Slope Equation
Now that we have the slope of the line, we can use the point-slope equation formula to find the point-slope equation of the line.
y - y1 = m(x - x1)
We can plug in the values of the two points (4, 1) and (2, -3) into the formula.
y - 1 = 2(x - 4)
This is the point-slope equation of the line.
Comparing the Answer Choices
Now that we have found the point-slope equation of the line, we can compare it to the answer choices.
A. y - 3 = 2(x - 2) B. y - 1 = 2(x - 4) C. y - 3 = -2(x + 2) D. y - 1 = -2(x - 4)
Our answer is B. y - 1 = 2(x - 4).
Conclusion
In this article, we learned how to find the point-slope equation of a line through two given points. We used the formula for the slope of a line and the point-slope equation formula to find the point-slope equation of the line. We also compared our answer to the answer choices and found that our answer is B. y - 1 = 2(x - 4).
Step-by-Step Solution
Here is the step-by-step solution to the problem:
- Find the slope of the line using the formula m = (y2 - y1) / (x2 - x1).
- Use the point-slope equation formula y - y1 = m(x - x1) to find the point-slope equation of the line.
- Plug in the values of the two points (4, 1) and (2, -3) into the formula.
- Simplify the equation to get the point-slope equation of the line.
- Compare the answer to the answer choices.
Practice Problems
Here are some practice problems to help you practice finding the point-slope equation of a line.
- Find the point-slope equation of the line through the points (2, 3) and (4, 5).
- Find the point-slope equation of the line through the points (1, 2) and (3, 4).
- Find the point-slope equation of the line through the points (5, 6) and (7, 8).
Answer Key
Here is the answer key to the practice problems.
- y - 3 = 1/2(x - 4)
- y - 2 = 1(x - 3)
- y - 6 = 1/2(x - 7)
Final Thoughts
Q: What is the point-slope equation of a line?
A: The point-slope equation of a line is a way to express the equation of a line using the coordinates of two points on the line. It is given by the formula:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, m is the slope of the line, and (x, y) is any point on the line.
Q: How do I find the slope of the line?
A: To find the slope of the line, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Q: What if the two points have the same x-coordinate?
A: If the two points have the same x-coordinate, then the line is vertical, and the slope is undefined. In this case, you can use the equation x = x1 to find the equation of the line.
Q: What if the two points have the same y-coordinate?
A: If the two points have the same y-coordinate, then the line is horizontal, and the slope is 0. In this case, you can use the equation y = y1 to find the equation of the line.
Q: How do I find the point-slope equation of a line when I know the slope and one point?
A: To find the point-slope equation of a line when you know the slope and one point, you can use the formula:
y - y1 = m(x - x1)
where (x1, y1) is the point you know, and m is the slope.
Q: Can I use the point-slope equation to find the equation of a line when I know two points and the slope is 0?
A: Yes, you can use the point-slope equation to find the equation of a line when you know two points and the slope is 0. In this case, the equation will be of the form y = y1.
Q: Can I use the point-slope equation to find the equation of a line when I know two points and the slope is undefined?
A: Yes, you can use the point-slope equation to find the equation of a line when you know two points and the slope is undefined. In this case, the equation will be of the form x = x1.
Q: How do I graph a line using the point-slope equation?
A: To graph a line using the point-slope equation, you can use the following steps:
- Find the slope of the line.
- Choose a point on the line.
- Use the point-slope equation to find the equation of the line.
- Graph the line using the equation.
Q: Can I use the point-slope equation to find the equation of a line when I know three points?
A: Yes, you can use the point-slope equation to find the equation of a line when you know three points. However, you will need to find the slope of the line first, and then use the point-slope equation to find the equation of the line.
Q: Can I use the point-slope equation to find the equation of a line when I know four points?
A: Yes, you can use the point-slope equation to find the equation of a line when you know four points. However, you will need to find the slope of the line first, and then use the point-slope equation to find the equation of the line.
Q: What if I make a mistake when finding the slope or the point-slope equation?
A: If you make a mistake when finding the slope or the point-slope equation, you may end up with an incorrect equation for the line. To avoid this, make sure to double-check your work and use a calculator or other tools to help you find the slope and the point-slope equation.
Q: Can I use the point-slope equation to find the equation of a line when I know the equation of a related line?
A: Yes, you can use the point-slope equation to find the equation of a line when you know the equation of a related line. For example, if you know the equation of a line and you want to find the equation of a line that is parallel to it, you can use the point-slope equation to find the equation of the new line.
Q: Can I use the point-slope equation to find the equation of a line when I know the equation of a related line and a point on the new line?
A: Yes, you can use the point-slope equation to find the equation of a line when you know the equation of a related line and a point on the new line. For example, if you know the equation of a line and you want to find the equation of a line that passes through a certain point, you can use the point-slope equation to find the equation of the new line.