Equation of a Line Passing Through a Point

Oct 13, 2025 by miodeya.com 57 views

Introduction

In mathematics, the equation of a line is a fundamental concept that is used to describe the relationship between two variables. The equation of a line can be expressed in various forms, including the slope-intercept form, point-slope form, and standard form. In this article, we will focus on the point-slope form and the standard form of the equation of a line.

Point-Slope Form

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

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope of the line.

Standard Form

The standard form of the equation of a line is given by:

Ax + By = C

where A, B, and C are constants.

Equation of a Line Passing Through a Point

Given a point (x1, y1) and a slope m, we can use the point-slope form to find the equation of the line passing through the point.

Example

Suppose we want to find the equation of the line passing through the point (-3, 2) and having a slope of 2/3.

Using the point-slope form, we can write:

y - 2 = (2/3)(x - (-3))

Simplifying the equation, we get:

y - 2 = (2/3)(x + 3)

Expanding the right-hand side, we get:

y - 2 = (2/3)x + 2

Adding 2 to both sides, we get:

y = (2/3)x + 4

Equations to Check

Now, let's check the given equations to see if they match the equation we derived.

A. 2x - 3y = -12

To check if this equation is correct, we can rewrite it in slope-intercept form.

Rearranging the terms, we get:

-3y = -2x - 12

Dividing both sides by -3, we get:

y = (2/3)x + 4

This equation matches the equation we derived earlier.

B. y = (2/3)x

This equation is already in slope-intercept form, and it matches the equation we derived earlier.

C. y = (2/3)x - 4

This equation is also in slope-intercept form, but it does not match the equation we derived earlier.

Conclusion

In conclusion, the correct equations for the line passing through the point (-3, 2) and having a slope of 2/3 are:

2x - 3y = -12
y = (2/3)x

These equations match the equation we derived using the point-slope form.

Key Takeaways

The point-slope form of the equation of a line is given by y - y1 = m(x - x1).
The standard form of the equation of a line is given by Ax + By = C.
To find the equation of a line passing through a point and having a slope, we can use the point-slope form.
We can check if an equation is correct by rewriting it in slope-intercept form and comparing it to the equation we derived.

Final Answer

The correct equations for the line passing through the point (-3, 2) and having a slope of 2/3 are:

2x - 3y = -12
y = (2/3)x
Equations of Lines: A Comprehensive Guide =====================================================

Q&A: Equations of Lines

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

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

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope of the line.

Q: What is the standard form of the equation of a line?

A: The standard form of the equation of a line is given by:

Ax + By = C

where A, B, and C are constants.

Q: How do I find the equation of a line passing through a point and having a slope?

A: To find the equation of a line passing through a point and having a slope, you can use the point-slope form. Simply substitute the values of the point and the slope into the equation:

y - y1 = m(x - x1)

Q: What is the slope-intercept form of the equation of a line?

A: The slope-intercept form of the equation of a line is given by:

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, simply solve for y:

y - y1 = m(x - x1)

Add y1 to both sides:

y = m(x - x1) + y1

Expand the right-hand side:

y = mx - mx1 + y1

Q: What is the difference between the point-slope form and the slope-intercept form?

A: The point-slope form is used to find the equation of a line passing through a point and having a slope, while the slope-intercept form is used to find the equation of a line with a given slope and y-intercept.

Q: Can I use the point-slope form to find the equation of a line with a given slope and y-intercept?

A: No, the point-slope form is used to find the equation of a line passing through a point and having a slope. If you want to find the equation of a line with a given slope and y-intercept, you should use the slope-intercept form.

Q: How do I check if an equation is correct?

A: To check if an equation is correct, you can rewrite it in slope-intercept form and compare it to the equation you derived using the point-slope form.

Q: What are some common mistakes to avoid when working with equations of lines?

A: Some common mistakes to avoid when working with equations of lines include:

Not using the correct form of the equation (point-slope or slope-intercept)
Not substituting the correct values into the equation
Not simplifying the equation correctly
Not checking if the equation is correct

Q: What are some real-world applications of equations of lines?

A: Equations of lines have many real-world applications, including:

Finding the equation of a line passing through two points
Finding the equation of a line with a given slope and y-intercept
Finding the equation of a line that passes through a point and has a given slope
Solving systems of linear equations

Conclusion

In conclusion, equations of lines are a fundamental concept in mathematics that have many real-world applications. By understanding the point-slope form and the slope-intercept form, you can find the equation of a line passing through a point and having a slope, and you can check if an equation is correct. Remember to avoid common mistakes and to use the correct form of the equation.