Finding the Equation of a Line Through Given Points

When dealing with problems in geometry and algebra, understanding the relationship between points and the equations of lines is fundamental. In this article, we delve into how to determine the equation of a line that passes through given points and satisfy certain conditions. Specifically, we explore how to find the equation of a line passing through the points (5, 0) and (0, 1).

Understanding the Given Problem

The problem statement mentions an equation of the line passing through the points (5, 0) and (0, 1) "legitimately totaling 1 point," which seems to be a mistranslation. The correct interpretation is to find the equation of the line that accurately passes through these two points.

We will start by calculating the slope using the two given points. The slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

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

Calculating the Slope

Using the points (5, 0) and (0, 1), we can calculate the slope as follows:

y2 - y1 1 - 0 1

x2 - x1 0 - 5 -5

slope (m) 1 / -5 -1/5

Deriving the Equation of the Line

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

y - y1 m(x - x1)

We can use either of the given points. Let's use the point (5, 0). Substituting the values, we get:

y - 0 -1/5(x - 5)

Simplifying this, we get:

y -1/5(x - 5)

Expanding and simplifying further:

y -1/5x 1

Verification

To verify that the equation y -1/5x 1 passes through the points (5, 0) and (0, 1), we substitute the coordinates of each point into the equation:

For point (5, 0):

0 -1/5(5) 1 -1 1 0

For point (0, 1):

1 -1/5(0) 1 0 1 1

Both points satisfy the equation, confirming that our derived equation is correct.

The Final Equation

The equation of the line passing through the points (5, 0) and (0, 1) is:

y -1/5x 1

Conclusion

By understanding the concept of slope and the point-slope form of a line, we can determine the equation of a line that passes through given points. In this case, we calculated the slope, used the point-slope form, and verified our work by substituting the points back into the equation. This is a fundamental skill in geometry and algebra, which has applications in various fields, including physics, engineering, and data analysis.

To summarize, the keys to solving this problem are:

Calculating the slope

Using the point-slope form to derive the equation

Verifying the solution by substituting the points