Point Through Which the Line Passes

Oct 13, 2025 by talynt.org 125 views

Introduction

In mathematics, the point-slope form of a linear equation is a powerful tool for describing lines on a coordinate plane. This form is particularly useful when we need to find the equation of a line that is parallel to a given line and passes through a specific point. In this article, we will explore how to identify an equation in point-slope form for a line that is parallel to a given line and passes through a given point.

Understanding Point-Slope Form

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

y - y1 = m(x - x1)

where (x1, y1) is a point on the line, and m is the slope of the line. This formula allows us to easily find the equation of a line that passes through a given point and has a given slope.

Parallel Lines

Two lines are said to be parallel if they have the same slope but different y-intercepts. In other words, if we have two lines with equations y = m1x + b1 and y = m2x + b2, then the lines are parallel if m1 = m2 and b1 ≠ b2.

Given Line

The given line is y = (3/4)x - 4. This line has a slope of 3/4 and a y-intercept of -4.

Point Through Which the Line Passes

The line passes through the point (-1, 7).

Finding the Equation of the Parallel Line

To find the equation of the parallel line, we need to find the slope of the line. Since the given line has a slope of 3/4, the parallel line will also have a slope of 3/4.

The point-slope form of the equation of the parallel line is:

y - 7 = (3/4)(x + 1)

This equation represents a line that passes through the point (-1, 7) and has a slope of 3/4.

Checking the Answer

Let's check the answer by plugging in the point (-1, 7) into the equation:

y - 7 = (3/4)(x + 1) 7 - 7 = (3/4)(-1 + 1) 0 = (3/4)(0) 0 = 0

The equation is true, so the answer is correct.

Conclusion

In this article, we learned how to identify an equation in point-slope form for a line that is parallel to a given line and passes through a given point. We used the point-slope form of a linear equation and the concept of parallel lines to find the equation of the parallel line. We also checked the answer by plugging in the point into the equation.

Final Answer

The final answer is:

A. $y - 7 = \frac{3}{4}(x + 1)$

This is the correct equation of the line that is parallel to the given line and passes through the given point.

Additional Examples

Here are some additional examples of finding the equation of a line that is parallel to a given line and passes through a given point:

Given line: y = 2x + 3 Point: (1, 5) Equation: y - 5 = 2(x - 1)
Given line: y = -x + 2 Point: (-2, 6) Equation: y - 6 = -1(x + 2)
Given line: y = 3x - 2 Point: (0, 1) Equation: y - 1 = 3(x - 0)

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

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

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 slope of a line?

A: The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

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

A: To find the equation of a line that is parallel to a given line and passes through a given point, you need to follow these steps:

Find the slope of the given line.
Use the point-slope form of a linear equation to write the equation of the line that passes through the given point and has the same slope as the given line.
Simplify the equation to get the final answer.

Q: What is the difference between a line and a plane?

A: A line is a one-dimensional object that extends infinitely in two directions, while a plane is a two-dimensional object that extends infinitely in all directions.

Q: Can a line be parallel to a plane?

A: No, a line cannot be parallel to a plane. A line is a one-dimensional object, while a plane is a two-dimensional object, so they cannot be parallel.

Q: Can a plane be parallel to a line?

A: Yes, a plane can be parallel to a line. In fact, a plane is parallel to a line if the line lies in the plane.

Q: What is the equation of a line that is parallel to the line y = 2x + 3 and passes through the point (1, 5)?

A: The equation of a line that is parallel to the line y = 2x + 3 and passes through the point (1, 5) is:

y - 5 = 2(x - 1)

Q: What is the equation of a line that is parallel to the line y = -x + 2 and passes through the point (-2, 6)?

A: The equation of a line that is parallel to the line y = -x + 2 and passes through the point (-2, 6) is:

y - 6 = -1(x + 2)

Q: What is the equation of a line that is parallel to the line y = 3x - 2 and passes through the point (0, 1)?

A: The equation of a line that is parallel to the line y = 3x - 2 and passes through the point (0, 1) is:

y - 1 = 3(x - 0)

Q: How do I check if the equation of a line is correct?

A: To check if the equation of a line is correct, you need to plug in the point into the equation and see if it is true. If the equation is true, then the equation is correct.

Q: What is the final answer to the problem of finding the equation of a line that is parallel to the line y = (3/4)x - 4 and passes through the point (-1, 7)?

A: The final answer is:

y - 7 = (3/4)(x + 1)

This is the correct equation of the line that is parallel to the given line and passes through the given point.