Understanding the Equation of a Line Through Given Points

When it comes to finding the equation of a line that passes through specific points, we often use the point-slope formula and the slope-intercept form. This article will walk you through the process of finding the equation of a line, specifically when given the points -3,4 and 7,-1. We will also explore a unique case where the x-coordinates of the points are the same.

General Method for Finding the Equation of a Line

The basic method to find the equation of a line passing through two points is to use the point-slope formula:

Point-Slope Formula

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

(frac{y - y_1}{y_2 - y_1} frac{x - x_1}{x_2 - x_1})

Given two points ((x_1, y_1)) and ((x_2, y_2)), we can substitute these points into the formula to find the equation of the line. For our points -3,4 and 7,-1:

(frac{y - 4}{-1 - 4} frac{x 3}{7 3})

This simplifies to:

(frac{y - 4}{-5} frac{x 3}{10})

By cross-multiplying and simplifying further, we can find the equation of the line:

(-10(y - 4) 5(x 3))

Which simplifies to:

(-10y 40 5x 15)

Or:

(-2y 8 x 3)

And further simplifying:

(x 2y - 5 0)

Special Case: Points with the Same x-Coordinate

When the x-coordinates of the points are the same, this means the line is vertical, and its equation is simply the x-coordinate of the points. For example, given the points -7,-3 and -7,1:

(x -7)

This case is special because the line is a vertical line, and its equation is always of the form:

(x a)

Where (a) is the common x-coordinate of the points.

Exploring the Line Equation y mx b

The slope-intercept form of a line is given by:

Slope-Intercept Form

(y mx b)

Where (m) is the slope and (b) is the y-intercept. However, when the x-coordinates are the same (as in the case of -7,-3 and -7,1), the slope is undefined, and the line cannot be expressed in slope-intercept form.

But we can express it in the form (L(x, y) 1) or (ax by 1) where (a) and (b) are constants. For the points -7,-3 and -7,1, we have:

(-3x - 7y 14 0)

Which simplifies to:

(-7x - 7y 0)

Or:

(x y 1)

Conclusion

In summary, finding the equation of a line through given points involves using the appropriate formula depending on the relationship between the points. When the x-coordinates are the same, the line is vertical and can be expressed as (x a). For other cases, the point-slope formula or slope-intercept form can be used, provided the slope is defined.