Write The Equation In Slope-intercept Form Of The Line That Passes Through The Given Points:11. ( − 1 , 3 (-1,3 ( − 1 , 3 ] And ( − 3 , 1 (-3,1 ( − 3 , 1 ]12. ( − 4 , 8 (-4,8 ( − 4 , 8 ] And ( 4 , 6 (4,6 ( 4 , 6 ]Write The Equation In Point-slope Form Of The Line That Passes Through The
Introduction
In mathematics, the slope-intercept and point-slope forms are two ways to represent a line on a coordinate plane. The slope-intercept form is a linear equation in the form of y = mx + b, where m is the slope and b is the y-intercept. The point-slope form, on the other hand, is a linear equation in the form of y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. In this article, we will explore how to write the equation of a line in slope-intercept and point-slope form using given points.
Slope-Intercept Form
The slope-intercept form of a line is a linear equation in the form of y = mx + b, where m is the slope and b is the y-intercept. To write the equation of a line in slope-intercept form, we need to find the slope and the y-intercept.
Finding the Slope
The slope of a line can be found using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Finding the Y-Intercept
Once we have the slope, we can find the y-intercept by substituting one of the points into the equation y = mx + b.
Example 1: Finding the Equation of a Line in Slope-Intercept Form
Given points: (-1, 3) and (-3, 1)
To find the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1) = (1 - 3) / (-3 - (-1)) = -2 / -2 = 1
Now that we have the slope, we can find the y-intercept by substituting one of the points into the equation y = mx + b.
Let's use the point (-1, 3):
3 = 1(-1) + b 3 = -1 + b b = 4
So, the equation of the line in slope-intercept form is:
y = x + 4
Example 2: Finding the Equation of a Line in Slope-Intercept Form
Given points: (-4, 8) and (4, 6)
To find the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1) = (6 - 8) / (4 - (-4)) = -2 / 8 = -1/4
Now that we have the slope, we can find the y-intercept by substituting one of the points into the equation y = mx + b.
Let's use the point (-4, 8):
8 = (-1/4)(-4) + b 8 = 1 + b b = 7
So, the equation of the line in slope-intercept form is:
y = (-1/4)x + 7
Point-Slope Form
The point-slope form of a line is a linear equation in the form of y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. To write the equation of a line in point-slope form, we need to find the slope and a point on the line.
Finding the Slope
The slope of a line can be found using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Finding a Point on the Line
We can use any point on the line to write the equation in point-slope form.
Example 1: Finding the Equation of a Line in Point-Slope Form
Given points: (-1, 3) and (-3, 1)
To find the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1) = (1 - 3) / (-3 - (-1)) = -2 / -2 = 1
Now that we have the slope, we can use one of the points to write the equation in point-slope form.
Let's use the point (-1, 3):
y - 3 = 1(x - (-1)) y - 3 = x + 1
Example 2: Finding the Equation of a Line in Point-Slope Form
Given points: (-4, 8) and (4, 6)
To find the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1) = (6 - 8) / (4 - (-4)) = -2 / 8 = -1/4
Now that we have the slope, we can use one of the points to write the equation in point-slope form.
Let's use the point (-4, 8):
y - 8 = (-1/4)(x - (-4)) y - 8 = (-1/4)(x + 4)
Conclusion
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is a linear equation in the form of y = mx + b, where m is the slope and b is the y-intercept.
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 two points on the line.
Q: How do I find the y-intercept of a line?
A: To find the y-intercept of a line, you can substitute one of the points into the equation y = mx + b.
Q: What is the point-slope form of a line?
A: The point-slope form of a line is a linear equation in the form of y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Q: How do I write the equation of a line in point-slope form?
A: To write the equation of a line in point-slope form, you need to find the slope and a point on the line. You can use the formula:
m = (y2 - y1) / (x2 - x1)
to find the slope, and then use one of the points to write the equation in point-slope form.
Q: What is the difference between slope-intercept and point-slope form?
A: The main difference between slope-intercept and point-slope form is that slope-intercept form uses the y-intercept (b) as the constant term, while point-slope form uses a point on the line (x1, y1) as the constant term.
Q: Can I use any point on the line to write the equation in point-slope form?
A: Yes, you can use any point on the line to write the equation in point-slope form. However, it's often easier to use a point that is given in the problem.
Q: How do I convert a point-slope equation to slope-intercept form?
A: To convert a point-slope equation to slope-intercept form, you can use the following steps:
- Distribute the slope (m) to the terms inside the parentheses.
- Simplify the equation.
- Move the constant term to the right-hand side of the equation.
Q: How do I convert a slope-intercept equation to point-slope form?
A: To convert a slope-intercept equation to point-slope form, you can use the following steps:
- Identify a point on the line (x1, y1).
- Substitute the point into the equation y = mx + b.
- Simplify the equation.
Q: What are some common mistakes to avoid when writing the equation of a line?
A: Some common mistakes to avoid when writing the equation of a line include:
- Forgetting to distribute the slope (m) to the terms inside the parentheses.
- Forgetting to simplify the equation.
- Using the wrong point on the line.
- Using the wrong slope (m).
Q: How do I check my work when writing the equation of a line?
A: To check your work when writing the equation of a line, you can use the following steps:
- Plug in the points given in the problem into the equation.
- Simplify the equation.
- Check that the equation is true for all points on the line.
By following these steps and avoiding common mistakes, you should be able to write the equation of a line in slope-intercept and point-slope form with confidence.