Which Equation Represents A Line That Passes Through \[$\left(4, \frac{1}{3}\right)\$\] And Has A Slope Of \[$\frac{3}{4}\$\]?A. \[$y - \frac{1}{3} = \frac{3}{4}(x - 4)\$\]B. \[$y - \frac{1}{3} = 4\left(x -

Feb 26, 2025 by ADMIN 207 views

Which Equation Represents a Line That Passes Through a Given Point and Has a Specific Slope?

In mathematics, the equation of a line can be represented in various forms, including the slope-intercept form, point-slope form, and standard form. When given a point that the line passes through and the slope of the line, we can use the point-slope form to find the equation of the line. In this article, we will explore how to find the equation of a line that passes through a given point and has a specific slope.

Understanding the Point-Slope Form

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 and m is the slope of the line. This form is useful when we are given a point that the line passes through and the slope of the line.

Example Problem

Let's consider the problem of finding the equation of a line that passes through the point (4, 1/3) and has a slope of 3/4. We can use the point-slope form to find the equation of the line.

Step 1: Identify the Given Information

The point that the line passes through is (4, 1/3).
The slope of the line is 3/4.

Step 2: Plug in the Values into the Point-Slope Form

We can plug in the values into the point-slope form as follows:

y - 1/3 = 3/4(x - 4)

Step 3: Simplify the Equation

We can simplify the equation by multiplying both sides by 12 to eliminate the fractions:

12(y - 1/3) = 12(3/4)(x - 4)

12y - 4 = 9(x - 4)

12y - 4 = 9x - 36

12y = 9x - 32

y = (9/12)x - 32/12

y = (3/4)x - 8/3

Step 4: Check the Answer Choices

We can check the answer choices to see which one matches the equation we found.

A. y - 1/3 = 3/4(x - 4)
B. y - 1/3 = 4(x - 4)
C. y - 1/3 = 3/4(x - 4)
D. y - 1/3 = 4(x - 4)

The correct answer is A. y - 1/3 = 3/4(x - 4).

In this article, we explored how to find the equation of a line that passes through a given point and has a specific slope. We used the point-slope form to find the equation of the line and checked the answer choices to see which one matched the equation we found. The correct answer is A. y - 1/3 = 3/4(x - 4).

Here are some additional examples of finding the equation of a line that passes through a given point and has a specific slope:

Find the equation of a line that passes through the point (2, 5) and has a slope of 2/3.
Find the equation of a line that passes through the point (3, 2) and has a slope of 1/2.
Find the equation of a line that passes through the point (1, 4) and has a slope of 3/2.

Here are some tips and tricks for finding the equation of a line that passes through a given point and has a specific slope:

Make sure to identify the given information correctly.
Plug in the values into the point-slope form correctly.
Simplify the equation correctly.
Check the answer choices to see which one matches the equation you found.

Here are some common mistakes to avoid when finding the equation of a line that passes through a given point and has a specific slope:

Not identifying the given information correctly.
Not plugging in the values into the point-slope form correctly.
Not simplifying the equation correctly.
Not checking the answer choices to see which one matches the equation you found.

Here are some real-world applications of finding the equation of a line that passes through a given point and has a specific slope:

Finding the equation of a line that represents a budget constraint.
Finding the equation of a line that represents a cost function.
Finding the equation of a line that represents a revenue function.

In conclusion, finding the equation of a line that passes through a given point and has a specific slope is an important concept in mathematics. We used the point-slope form to find the equation of the line and checked the answer choices to see which one matched the equation we found. The correct answer is A. y - 1/3 = 3/4(x - 4). We also provided additional examples, tips and tricks, and common mistakes to avoid.
Frequently Asked Questions (FAQs) About Finding the Equation of a Line

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 and m is the slope of the line.

Q: How do I find the equation of a line that passes through a given point and has a specific slope?

A: To find the equation of a line that passes through a given point and has a specific slope, you can use the point-slope form. Plug in the values of the point and the slope into the equation, and simplify to find the equation of 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 simplify the equation by isolating y. This will give you the equation in the slope-intercept form.

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 two-point form. This involves finding the slope of the line using the two points, and then using the point-slope form to find the equation of the line.

Q: What is the two-point form of a line?

A: The two-point form of a line is given by the equation:

y - y1 = m(x - x1)

where (x1, y1) and (x2, y2) are two points on the line, and m is the slope of the line.

Q: How do I find the equation of a line that is parallel to a given line?

A: To find the equation of a line that is parallel to a given line, you can use the fact that parallel lines have the same slope. Find the slope of the given line, and then use the point-slope form to find the equation of the line.

Q: How do I find the equation of a line that is perpendicular to a given line?

A: To find the equation of a line that is perpendicular to a given line, you can use the fact that perpendicular lines have slopes that are negative reciprocals of each other. Find the slope of the given line, and then find the negative reciprocal of the slope. Use the point-slope form to find the equation of the line.

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 identifying the given information correctly.
Not plugging in the values into the point-slope form correctly.
Not simplifying the equation correctly.
Not checking the answer choices to see which one matches the equation you found.

Q: What are some real-world applications of finding the equation of a line?

A: Some real-world applications of finding the equation of a line include:

Finding the equation of a line that represents a budget constraint.
Finding the equation of a line that represents a cost function.
Finding the equation of a line that represents a revenue function.

In conclusion, finding the equation of a line that passes through a given point and has a specific slope is an important concept in mathematics. We answered some frequently asked questions about finding the equation of a line, including the point-slope form, the slope-intercept form, and the standard form. We also discussed how to find the equation of a line that passes through two points, and how to find the equation of a line that is parallel or perpendicular to a given line.