What Is The Equation Of The Line Containing The Points $(3,1),(9,3),$ And $(27,9$\]?A. $y=3x$ B. $y=x^3$ C. $y=\frac{1}{3}x$
Introduction
In mathematics, finding the equation of a line that passes through multiple points is a fundamental concept in algebra and geometry. Given three points, we can determine the equation of the line that contains them. In this article, we will explore how to find the equation of the line containing the points (3,1), (9,3), and (27,9).
Understanding the Problem
To find the equation of the line, we need to understand the relationship between the points and the line. The points (3,1), (9,3), and (27,9) are given, and we need to find the equation of the line that passes through these points. This means that the line must contain all three points.
Finding the Slope of the Line
The slope of a line is a measure of how steep it is. It can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where m is the slope, and (x1, y1) and (x2, y2) are two points on the line.
Using the given points (3,1) and (9,3), we can calculate the slope of the line:
m = (3 - 1) / (9 - 3) m = 2 / 6 m = 1/3
Finding the Equation of the Line
Now that we have the slope of the line, we can use it to find the equation of the line. The equation of a line can be written in the form:
y = mx + b
where m is the slope, and b is the y-intercept.
We can use the point (3,1) to find the value of b:
1 = (1/3)(3) + b 1 = 1 + b b = 0
Writing the Equation of the Line
Now that we have the slope and the y-intercept, we can write the equation of the line:
y = (1/3)x + 0 y = (1/3)x
Conclusion
In this article, we have found the equation of the line containing the points (3,1), (9,3), and (27,9). The equation of the line is y = (1/3)x. This means that for every value of x, the corresponding value of y is one-third of x.
Comparison with the Given Options
Let's compare our answer with the given options:
A. y = 3x B. y = x^3 C. y = (1/3)x
Our answer, y = (1/3)x, matches option C.
Final Answer
The final answer is C. y = (1/3)x.
Introduction
In our previous article, we explored how to find the equation of a line containing multiple points. We used the points (3,1), (9,3), and (27,9) to find the equation of the line. In this article, we will answer some frequently asked questions related to finding the equation of a line containing multiple points.
Q: What is the equation of a line containing the points (4,2), (8,4), and (12,6)?
A: To find the equation of the line, we need to calculate the slope using the formula:
m = (y2 - y1) / (x2 - x1)
Using the points (4,2) and (8,4), we get:
m = (4 - 2) / (8 - 4) m = 2 / 4 m = 1/2
Now, we can use the point (4,2) to find the value of b:
2 = (1/2)(4) + b 2 = 2 + b b = 0
The equation of the line is:
y = (1/2)x + 0 y = (1/2)x
Q: How do I find the equation of a line containing three points that are not in order?
A: To find the equation of a line containing three points that are not in order, you can use the following steps:
- Calculate the slope using the formula:
m = (y2 - y1) / (x2 - x1)
-
Use the point with the smallest x-value to find the value of b.
-
Write the equation of the line in the form:
y = mx + b
Q: What is the equation of a line containing the points (1,3), (2,5), and (3,7)?
A: To find the equation of the line, we need to calculate the slope using the formula:
m = (y2 - y1) / (x2 - x1)
Using the points (1,3) and (2,5), we get:
m = (5 - 3) / (2 - 1) m = 2 / 1 m = 2
Now, we can use the point (1,3) to find the value of b:
3 = (2)(1) + b 3 = 2 + b b = 1
The equation of the line is:
y = 2x + 1
Q: How do I know if three points are collinear?
A: Three points are collinear if they lie on the same line. To check if three points are collinear, you can use the following steps:
- Calculate the slope using the formula:
m = (y2 - y1) / (x2 - x1)
-
Use the point with the smallest x-value to find the value of b.
-
Write the equation of the line in the form:
y = mx + b
- Check if the equation of the line is the same for all three points.
If the equation of the line is the same for all three points, then the points are collinear.
Q: What is the equation of a line containing the points (0,2), (2,4), and (4,6)?
A: To find the equation of the line, we need to calculate the slope using the formula:
m = (y2 - y1) / (x2 - x1)
Using the points (0,2) and (2,4), we get:
m = (4 - 2) / (2 - 0) m = 2 / 2 m = 1
Now, we can use the point (0,2) to find the value of b:
2 = (1)(0) + b 2 = b
The equation of the line is:
y = x + 2
Conclusion
In this article, we have answered some frequently asked questions related to finding the equation of a line containing multiple points. We have provided examples and step-by-step solutions to help you understand the concept better.