Understanding the Line Equation Through Points: A Comprehensive Guide

When working with lines in the x-y plane, it's essential to understand how to determine the equation of a line that passes through specific points. This article will guide you through the process of finding the equation of a line that goes through the points (1, 3) and (3, 5), explaining the fundamental concepts of slope and the point-slope form of the equation.

Introduction to the Slope-Intercept Form

The slope-intercept form of a line is expressed as y mx c, where m is the slope of the line and c is the y-intercept (the point where the line crosses the y-axis).

Solving for the Slope

To find the slope of a line passing through two points, (x1, y1) and (x2, y2), we use the slope formula:

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

Consider the points (1, 3) and (3, 5). Plugging these values into the formula, we get:

m (5 - 3) / (3 - 1) 2 / 2 1

Therefore, the slope m is 1. However, there seems to be a misunderstanding in the problem statement or a need to clarify. Let's calculate step-by-step as if the x2 was 2, which is a common oversight:

m (5 - 3) / (3 - 1) 2 / 2 1

So, the slope m is actually 2. Let's correct this and proceed with m 2.

Using the Point-Slope Form

Once we have the slope, we can use the point-slope form of the equation of a line, which is:

y - y1 m(x - x1)

Using the point (1, 3) and the slope m 2, we plug these values into the formula:

y - 3 2(x - 1)

Now, let's simplify this equation to get it into the slope-intercept form:

y - 3 2x - 2

y 2x - 2 3

y 2x 1

Thus, the equation of the line passing through the points (1, 3) and (3, 5) is:

y 2x - 1

Additional Considerations

Understanding the line equation through points is crucial in many fields, including geometry, physics, and engineering. The slope-intercept form provides a clear and concise way to describe a line. Furthermore, knowing the point-slope form allows for flexibility in using different points, making the equation-solving process more straightforward.

In conclusion, the equation of the line passing through the points (1, 3) and (3, 5) is y 2x - 1. This line has a slope of 2 and crosses the y-axis at -1. By mastering these concepts, you can confidently solve similar problems and apply the knowledge in various real-world scenarios.