Choose The Equation That Represents A Line That Passes Through The Points \[$(-1, 2)\$\] And \[$ (3, 1) \$\].A. \[$4x - Y = -6\$\]B. \[$x + 4y = 7\$\]C. \[$x - 4y = -9\$\]D. \[$4x + Y = 2\$\]
Introduction
In mathematics, a linear equation is an equation in which the highest power of the variable(s) is 1. It is a fundamental concept in algebra and is used to represent a line on a coordinate plane. In this article, we will discuss how to find the equation of a line that passes through two given points.
What is a Linear Equation?
A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are variables.
Finding the Equation of a Line
To find the equation of a line that passes through two given points, we can use the following steps:
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Find the slope of the line: The slope of a line is a measure of how steep it is. It can be found using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
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Use the point-slope form of a line: The point-slope form of a line is:
y - y1 = m(x - x1)
where (x1, y1) is one of the points and m is the slope.
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Simplify the equation: Once we have the point-slope form of the line, we can simplify it to find the equation of the line in the form ax + by = c.
Example Problem
Let's consider the problem of finding the equation of a line that passes through the points (-1, 2) and (3, 1).
Step 1: Find the slope of the line
To find the slope of the line, we can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (-1, 2) and (x2, y2) = (3, 1).
Plugging in the values, we get:
m = (1 - 2) / (3 - (-1)) m = -1 / 4 m = -0.25
Step 2: Use the point-slope form of a line
Now that we have the slope, we can use the point-slope form of a line:
y - y1 = m(x - x1)
where (x1, y1) = (-1, 2) and m = -0.25.
Plugging in the values, we get:
y - 2 = -0.25(x - (-1)) y - 2 = -0.25(x + 1) y - 2 = -0.25x - 0.25
Step 3: Simplify the equation
To simplify the equation, we can add 2 to both sides:
y = -0.25x - 0.25 + 2 y = -0.25x + 1.75
Now that we have the equation of the line in the form y = mx + b, we can rewrite it in the form ax + by = c by multiplying both sides by -4:
4y = -x + 7
4x - y = -6
x + 4y = 7
x - 4y = -9
4x + y = 2
Conclusion
In this article, we discussed how to find the equation of a line that passes through two given points. We used the point-slope form of a line and simplified the equation to find the equation of the line in the form ax + by = c. We also considered an example problem to illustrate the steps involved in finding the equation of a line.
Final Answer
The correct answer is:
A. 4x - y = -6
Introduction
In our previous article, we discussed how to find the equation of a line that passes through two given points. In this article, we will provide a Q&A guide to help you understand the concepts and solve linear equations.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are variables.
Q: How do I find the equation of a line that passes through two given points?
A: To find the equation of a line that passes through two given points, you can use the following steps:
-
Find the slope of the line: The slope of a line is a measure of how steep it is. It can be found using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
-
Use the point-slope form of a line: The point-slope form of a line is:
y - y1 = m(x - x1)
where (x1, y1) is one of the points and m is the slope.
-
Simplify the equation: Once you have the point-slope form of the line, you can simplify it to find the equation of the line in the form ax + by = c.
Q: What is the point-slope form of a line?
A: The point-slope form of a line is:
y - y1 = m(x - x1)
where (x1, y1) is one of the points and m is the slope.
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 the coordinates of the two points.
Q: What is the equation of a line in the form ax + by = c?
A: The equation of a line in the form ax + by = c is a linear equation in which the highest power of the variable(s) is 1.
Q: How do I simplify the equation of a line?
A: To simplify the equation of a line, you can add or subtract the same value to both sides of the equation.
Q: What is the final answer to the example problem?
A: The final answer to the example problem is:
A. 4x - y = -6
This is the equation of the line that passes through the points (-1, 2) and (3, 1).
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when solving linear equations.
- Not simplifying the equation: Make sure to simplify the equation to find the final answer.
- Not checking the solution: Make sure to check the solution to ensure that it is correct.
Conclusion
In this article, we provided a Q&A guide to help you understand the concepts and solve linear equations. We discussed the point-slope form of a line, how to find the slope of a line, and how to simplify the equation of a line. We also provided some common mistakes to avoid when solving linear equations.
Final Tips
- Practice, practice, practice: The more you practice solving linear equations, the more comfortable you will become with the concepts.
- Use online resources: There are many online resources available to help you learn and practice solving linear equations.
- Seek help when needed: Don't be afraid to seek help when you need it. Ask your teacher or tutor for assistance, or seek help from online resources.