Understanding the Y-Intercept on a Straight Line: A Detailed Guide

In the context of linear equations, the y-intercept is a crucial concept that refers to the point where a line crosses the y-axis. This article will explore how to find the y-intercept, why it's significant, and how to determine if points are collinear using the y-intercept and slope.

What is the Y-Intercept?

The y-intercept is the y-coordinate where a line intersects the y-axis. It's the point where the value of x is zero. When a point is given as (0, y), this y-coordinate is the y-intercept. In the given example, the point (0, 5) is provided, indicating that the y-intercept is 5.

Finding the Slope of a Line

The slope of a line is defined as the change in y divided by the change in x between any two points on the line. It can be calculated using the formula:

slope (y2 - y1) / (x2 - x1)

Given the points on a straight line in the problem as (38, 23) and (32, 8), the slope can be determined as follows:

slope (8 - 3) / (32 - 38) 5 / -6 -5/6 ≈ -0.833

However, the problem indicates that the slope is 1. This discrepancy needs to be revisited for clarity.

Understanding the Equation of the Line

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

y - y1 m(x - x1)

Given a point (x1, y1) and the slope m, this equation can be used to determine the equation of the line. For the point (2, 3) and slope 1:

y - 3 1(x - 2) y - 3 x - 2 y x 1

The point (0, 5) satisfies this equation, as:

y 0 1 5

Thus, the y-intercept is 5 at x 0.

Proving Points are Collinear via the Y-Intercept

To check if three points are collinear, you can use the concept of the y-intercept and calculate the slopes between the points. If the slopes between all pairs of points are equal, the points are collinear.

For the points (0, -159), (16, -10), and (9, -4), we calculate the slopes:

1. Slope between (0, -159) and (16, -10):

m1 (-10 - (-159)) / (16 - 0) 149 / 16 ≈ 9.3125

2. Slope between (16, -10) and (9, -4):

m2 (-4 - (-10)) / (9 - 16) 6 / -7 ≈ -0.857

Since the slopes are not equal, the points are not collinear.

The shoelace method is also used to confirm collinearity. The determinant of the shoelace formula 'area' is calculated as:

0(10) 16(-4) 9(-159) - (16(-10) 9(-159) 0(-4)) 0 - 64 - 1431 - (-160 - 1431 0) -1495 1591 96

Since the determinant is not zero, the points are not collinear.

Conclusion

The y-intercept is a fundamental concept in linear equations, providing insights into the point where a line crosses the y-axis. In this example, the y-intercept is 5, as shown by the point (0, 5) on the line with a slope of 1. Understanding the y-intercept and how to use it in conjunction with the slope and other points helps in determining the collinearity of points.

For more detailed resources on this topic, consider exploring additional articles and videos focusing on linear equations, slopes, and collinearity.