Which Equation Represents The Line That Passes Through The Points { (-6, 7)$}$ And { (-3, 6)$}$?A. { Y = -\frac{1}{3}x + 5$}$ B. { Y = -\frac{1}{3}x + 9$}$ C. { Y = -3x - 11$}$ D. [$y = -3x +
Introduction
In mathematics, the equation of a line can be represented in various forms, including the slope-intercept form, point-slope form, and standard form. To find the equation of a line that passes through two points, we can use the point-slope form, which is given by the equation:
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.
Step 1: Find the Slope of the Line
To find the slope of the line that passes through the points (-6, 7) and (-3, 6), we can use the slope formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (-6, 7) and (x2, y2) = (-3, 6).
Plugging in the values, we get:
m = (6 - 7) / (-3 - (-6)) m = -1 / 3 m = -1/3
Step 2: Use the Point-Slope Form to Find the Equation of the Line
Now that we have the slope of the line, we can use the point-slope form to find the equation of the line. We will use the point (-6, 7) as the point (x1, y1).
y - 7 = (-1/3)(x - (-6)) y - 7 = (-1/3)(x + 6) y - 7 = (-1/3)x - 2 y = (-1/3)x + 5
Step 3: Check the Answer Choices
Now that we have the equation of the line, we can check the answer choices to see which one matches our equation.
A. y = -1/3x + 5 B. y = -1/3x + 9 C. y = -3x - 11 D. y = -3x + 5
Our equation matches answer choice A.
Conclusion
In this article, we learned how to find the equation of a line that passes through two points. We used the point-slope form to find the equation of the line, and we checked the answer choices to see which one matched our equation. The correct answer is A. y = -1/3x + 5.
Additional Examples
Here are a few more examples of finding the equation of a line that passes through two points.
Example 1
Find the equation of the line that passes through the points (2, 3) and (4, 5).
Step 1: Find the Slope of the Line
m = (5 - 3) / (4 - 2) m = 2 / 2 m = 1
Step 2: Use the Point-Slope Form to Find the Equation of the Line
y - 3 = 1(x - 2) y - 3 = x - 2 y = x + 1
Example 2
Find the equation of the line that passes through the points (-2, 4) and (1, 2).
Step 1: Find the Slope of the Line
m = (2 - 4) / (1 - (-2)) m = -2 / 3 m = -2/3
Step 2: Use the Point-Slope Form to Find the Equation of the Line
y - 4 = (-2/3)(x - (-2)) y - 4 = (-2/3)(x + 2) y - 4 = (-2/3)x - 4/3 y = (-2/3)x + 8/3
Example 3
Find the equation of the line that passes through the points (3, 2) and (5, 4).
Step 1: Find the Slope of the Line
m = (4 - 2) / (5 - 3) m = 2 / 2 m = 1
Step 2: Use the Point-Slope Form to Find the Equation of the Line
y - 2 = 1(x - 3) y - 2 = x - 3 y = x - 1
Conclusion
Q: What is the point-slope form of a line?
A: The point-slope form of a line is given by the equation:
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 a line?
A: To find the slope of a line, you can use the slope formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is given by the equation:
y = mx + b
where m is the slope of the line and b is the y-intercept.
Q: How do I convert the point-slope form to the slope-intercept form?
A: To convert the point-slope form to the slope-intercept form, you can use the following steps:
- Simplify the equation by combining like terms.
- Isolate the y variable by subtracting the x term from both sides.
- Factor out the slope (m) from the equation.
Q: What is the standard form of a line?
A: The standard form of a line is given by the equation:
Ax + By = C
where A, B, and C are constants.
Q: How do I find the equation of a line that passes through two points?
A: To find the equation of a line that passes through two points, you can use the point-slope form and follow these steps:
- Find the slope of the line using the slope formula.
- Use the point-slope form to write the equation of the line.
- Simplify the equation by combining like terms.
Q: What are some common mistakes to avoid when finding the equation of a line?
A: Some common mistakes to avoid when finding the equation of a line include:
- Not using the correct formula for the slope.
- Not simplifying the equation correctly.
- Not checking the answer choices to see which one matches the equation.
Q: How do I check my answer to see if it is correct?
A: To check your answer, you can use the following steps:
- Plug in the values of x and y into the equation to see if it is true.
- Check the answer choices to see which one matches the equation.
- Use a graphing calculator or a graphing tool to visualize the line and see if it passes through the two points.
Conclusion
In this article, we answered some frequently asked questions about finding the equation of a line. We covered topics such as the point-slope form, slope-intercept form, standard form, and common mistakes to avoid. We also provided some tips on how to check your answer to see if it is correct.