Understanding the Number of Lines Passing Through Multiple Points

The question of determining the number of straight lines that can pass through a given set of points often involves an exploration of geometric principles and algebraic techniques. In this article, we will delve into the specifics of the provided set of points and determine how many lines can pass through them.

The Given Points and the Line Question

Consider the points: 1, 3, 7, -3, 5, -1, 6, -2. The focus is on understanding if multiple lines can pass through these points and how to calculate this.

General Concept: Number of Lines Through Points

When dealing with a set of n points, the maximum number of lines that can be formed by choosing two points from the set is given by the combination formula C(n, 2) n! / [2! * (n - 2)!]. This indicates that from a set of 8 points, the maximum number of unique lines that can be formed is C(8, 2) 28.

Specific Case: Points on the Same Line

However, in the specific case where the points are aligned in a straight line, the number of lines that can pass through them is significantly reduced. The points provided are (1, 3), (7, -3), (5, -1), (6, -2). The question here is whether these points lie on a unique line and, if so, find that line.

Determining the Unique Line

The strategy to solve this problem involves calculating the slope and using the point-slope form to find the line equation. If all the points lie on the same line, they will satisfy the same linear equation.

Calculating the Slope

The slope m of the line passing through two points (x1, y1) and (x2, y2) is given by:
m (y2 - y1) / (x2 - x1)

Let's take the points (1, 3) and (7, -3) to find the slope:

m (-3 - 3) / (7 - 1) -6 / 6 -1

Using the Slope-Intercept Form

The equation of a line in slope-intercept form is y mx b. We can use the slope -1 and one of the points, say (1, 3), to find the y-intercept b by substituting into the equation:

3 -1 * 1 b
b 3 1 4

Therefore, the equation of the line passing through these points is:

y -x 4 or equivalently in standard form:

x y 4

Verification of Other Points

To verify if the line x y 4 also passes through the remaining points, we can substitute them into the equation:

(7, -3) -> 7 (-3) 4 (5, -1) -> 5 (-1) 4 (6, -2) -> 6 (-2) 4

Since all the points satisfy the equation x y 4, we can conclude that only one unique line passes through all the given points.

Conclusion

Through this analysis, we have determined that the points (1, 3), (7, -3), (5, -1), (6, -2) lie on a single straight line, which is represented by the equation x y 4. This confirms that there is only one line passing through all four points, fulfilling the geometric principle that a unique line can be defined by any two points and that no other line can pass through all these points simultaneously.

Understanding such geometric problems provides a solid foundation in algebra and geometry, crucial for students and professionals in mathematics, engineering, and related fields. For more questions or to explore similar topics, visit our resources section for additional explanations and practice problems.