Understanding whether a given point lies on a specific line can be a crucial aspect of many mathematical and geometrical problems. In this guide, we will explore various methods and techniques to determine if a point belongs to a line. Let’s dive in!

What is a point?

A point is the most fundamental object in geometry. It has no size or shape and is represented by a dot. In a Cartesian coordinate system, a point is defined by its coordinates, usually denoted as (x, y).

What is a line?

A line is a straight path that extends infinitely in both directions. It is determined by two distinct points on the line, and any additional point on the line can be expressed as a linear combination of these two points.

How to determine if a point lies on a line?

There are two primary methods to determine if a given point belongs to a line:

  • Slope-intercept form: The slope-intercept form of a line is y = mx + b, where m represents the slope and b represents the y-intercept. To check if a point (x, y) lies on this line, substitute the x and y values into the equation. If the equation holds true, the point lies on the line!
  • Point-slope form: The point-slope form of a line is y – y₁ = m(x – x₁), where (x₁, y₁) represents a point on the line and m represents the slope. Plug in the coordinates of the given point and check if the equation holds true. If it does, the point lies on the line!

Examples:

Let’s work through a couple of examples to solidify our understanding.

Example 1:

Given the line equation y = 2x + 3 and a point (4, 11), does the point lie on the line?

Slope-intercept form:

Substitute x = 4 and y = 11 into the equation:

11 = 2(4) + 3

11 = 8 + 3

11 = 11

Since the equation holds true, the point (4, 11) lies on the line y = 2x + 3.

Example 2:

Given the line passing through points (2, 5) and (6, 11), does the point (3, 7) belong to the line?

Point-slope form:

Substitute x₁ = 2, y₁ = 5, m = (11 – 5) / (6 – 2) = 6 / 4 = 1.5 into the equation:

y – 5 = 1.5(x – 2)

Plugging in x = 3, y = 7:

7 – 5 = 1.5(3 – 2)

2 = 1.5

Since the equation doesn’t hold true, the point (3, 7) does not lie on the line passing through (2, 5) and (6, 11).

In Conclusion

Determining if a given point belongs to a line is an essential skill in various mathematical and geometrical contexts. By understanding the slope-intercept form and the point-slope form, you can easily test whether a point lies on a given line. Remember to substitute the point’s coordinates into the equation and check for equality. Practice these techniques, and you’ll master this fundamental aspect of geometry in no time!

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