Equation Of A Line Through (9,4) And (-6,-8)

Introduction

In mathematics, determining the equation of a line that passes through two given points is a fundamental concept in coordinate geometry. This process involves finding the slope of the line and then using either the point-slope form or the slope-intercept form to express the equation. In this article, we will walk through the steps to find the equation of a line that passes through the points $(9, 4)$ and $(-6, -8)$, and express the answer in fully simplified point-slope form, unless it is a vertical or horizontal line.

Step 1: Calculate the Slope

The slope of a line, often denoted as m, measures the steepness and direction of the line. Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope m can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

In our case, the given points are $(9, 4)$ and $(-6, -8)$. Let’s plug these values into the formula:

$$m = \frac{-8 - 4}{-6 - 9} = \frac{-12}{-15} = \frac{4}{5}$$

So, the slope of the line passing through the points $(9, 4)$ and $(-6, -8)$ is $\frac{4}{5}$. This positive slope indicates that the line is increasing from left to right. Understanding the slope is crucial because it tells us how much the y-value changes for every unit change in the x-value. A larger slope means a steeper line, while a smaller slope means a flatter line. A zero slope indicates a horizontal line, and an undefined slope indicates a vertical line. With the slope calculated, we can now proceed to the next step: using the point-slope form to write the equation of the line.

Step 2: Use the Point-Slope Form

The point-slope form of a linear equation is a convenient way to express the equation of a line when you know a point on the line and the slope. The point-slope form is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line and m is the slope of the line. We have already calculated the slope m to be $\frac{4}{5}$. Now, we can use either of the given points to plug into the point-slope form. Let’s use the point $(9, 4)$. Plugging these values into the point-slope form, we get:

$$y - 4 = \frac{4}{5}(x - 9)$$

This equation represents the line that passes through the points $(9, 4)$ and $(-6, -8)$. The point-slope form is particularly useful because it directly incorporates the slope and a point on the line, making it easy to write the equation without further manipulation. However, it’s important to note that this form is not unique, as using the other point $(-6, -8)$ would yield a different, but equivalent, equation. The point-slope form provides a flexible way to represent linear equations, especially when dealing with specific points and slopes. In the next step, we will simplify this equation to ensure it is in its fully simplified form.

Step 3: Simplify the Equation (Optional, but Recommended)

The equation we derived in the point-slope form is:

$$y - 4 = \frac{4}{5}(x - 9)$$

This equation is already in a simplified form of the point-slope equation. While further simplification can be done to convert it into slope-intercept form ($y = mx + b$), the question specifically asks for the answer in point-slope form. Therefore, no further simplification is needed. The point-slope form is useful because it directly shows the slope and a point on the line. If you were to convert it to slope-intercept form, you would distribute the $\frac{4}{5}$ and then isolate y.

$$y - 4 = \frac{4}{5}x - \frac{36}{5}$$

$$y = \frac{4}{5}x - \frac{36}{5} + 4$$

$$y = \frac{4}{5}x - \frac{36}{5} + \frac{20}{5}$$

$$y = \frac{4}{5}x - \frac{16}{5}$$

However, since we need to keep it in point-slope form, our final answer remains:

$$y - 4 = \frac{4}{5}(x - 9)$$

This is the fully simplified point-slope form of the line that passes through the points $(9, 4)$ and $(-6, -8)$.

Alternative Point-Slope Form

As mentioned earlier, the point-slope form is not unique because you can use either of the given points. Let’s verify this by using the point $(-6, -8)$ in the point-slope form. The equation becomes:

$$y - (-8) = \frac{4}{5}(x - (-6))$$

$$y + 8 = \frac{4}{5}(x + 6)$$

This is another valid point-slope form of the same line. To confirm that both equations represent the same line, you can convert both to the slope-intercept form and see if they match. We already converted the first point-slope form to slope-intercept form:

$$y = \frac{4}{5}x - \frac{16}{5}$$

Now, let’s convert the second point-slope form:

$$y + 8 = \frac{4}{5}(x + 6)$$

$$y + 8 = \frac{4}{5}x + \frac{24}{5}$$

$$y = \frac{4}{5}x + \frac{24}{5} - 8$$

$$y = \frac{4}{5}x + \frac{24}{5} - \frac{40}{5}$$

$$y = \frac{4}{5}x - \frac{16}{5}$$

Both point-slope forms convert to the same slope-intercept form, confirming that they represent the same line. Therefore, either form is a correct representation in point-slope form.

Conclusion

In summary, finding the equation of a line that passes through two given points involves calculating the slope and then using the point-slope form. We found that the slope of the line passing through $(9, 4)$ and $(-6, -8)$ is $\frac{4}{5}$. Using the point $(9, 4)$, we expressed the equation in point-slope form as:

$$y - 4 = \frac{4}{5}(x - 9)$$

Alternatively, using the point $(-6, -8)$, we found another valid point-slope form:

$$y + 8 = \frac{4}{5}(x + 6)$$

Both equations represent the same line and are correct in point-slope form. This exercise demonstrates the application of coordinate geometry principles to determine the equation of a line, a fundamental skill in mathematics.

Understanding these concepts allows you, guys, to easily solve similar problems and apply them in various mathematical and real-world contexts. Whether you're working on a homework assignment or analyzing data, knowing how to find the equation of a line is a valuable tool. Keep practicing, and you'll master it in no time! Remember, math can be fun and rewarding when you understand the underlying principles.