Linear Function From Point-Slope Equation Y + 7 = -2/3(x + 6)

In the realm of mathematics, linear functions stand as fundamental building blocks, shaping our understanding of relationships between variables. One common way to represent these functions is through the point-slope equation, a powerful tool that allows us to define a line using a single point and its slope. But what happens when we need to express this line in a more familiar form, such as the slope-intercept form? Let's embark on a journey to unravel this transformation, exploring the intricacies of linear functions and the elegance of algebraic manipulation. This article will serve as your comprehensive guide to navigate the conversion process, equipping you with the skills to confidently tackle similar problems. We'll delve into the core concepts, dissect the equation, and meticulously walk through the steps required to unveil the linear function lurking within the point-slope representation. So, buckle up and prepare to immerse yourself in the fascinating world of linear equations!

Understanding the Point-Slope Form

The point-slope form of a linear equation is a powerful way to represent a line when you know a specific point on the line and its slope. This form is expressed as:

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

where:

• $(x_1, y_1)$ represents a known point on the line.
• $m$ represents the slope of the line.

The point-slope form provides a direct way to construct the equation of a line when you have this information readily available. It highlights the relationship between the slope, a specific point, and the general variables x and y that define any point on the line. Understanding this form is crucial for manipulating and converting linear equations into different representations, such as the slope-intercept form, which we will explore in detail later. The beauty of the point-slope form lies in its ability to capture the essence of a line's behavior with just two key pieces of information: a point and its slope. It's a concise and efficient way to describe a line's direction and position within the coordinate plane.

In the given equation, $y + 7 = -\frac{2}{3}(x + 6)$, we can directly identify the point and slope. By comparing it to the general form, we see that:

• $y - y_1$ corresponds to $y + 7$, which can be rewritten as $y - (-7)$. Therefore, $y_1 = -7$.
• $x - x_1$ corresponds to $x + 6$, which can be rewritten as $x - (-6)$. Therefore, $x_1 = -6$.
• $m$ corresponds to $-\frac{2}{3}$, which is the slope of the line.

Thus, the point on the line is $(-6, -7)$ and the slope is $-\frac{2}{3}$. This initial decoding of the point-slope form is a fundamental step in our journey to unveil the linear function in its more familiar slope-intercept guise. It lays the groundwork for the algebraic manipulations that will follow.

Transforming to Slope-Intercept Form

The slope-intercept form is another common way to represent a linear equation, and it's expressed as:

$y = mx + b$

where:

• $m$ represents the slope of the line (the same m as in the point-slope form).
• $b$ represents the y-intercept, which is the point where the line crosses the y-axis.

The slope-intercept form is particularly useful because it explicitly shows the slope and y-intercept, making it easy to visualize the line and understand its behavior. The transformation from point-slope form to slope-intercept form involves a series of algebraic manipulations aimed at isolating y on one side of the equation. This process allows us to rewrite the equation in a form that directly reveals the slope and y-intercept.

To convert the given equation, $y + 7 = -\frac{2}{3}(x + 6)$, to slope-intercept form, we need to follow these steps:

1. Distribute the slope $-\frac{2}{3}$ across the terms inside the parentheses:

   $y + 7 = -\frac{2}{3}x - \frac{2}{3}(6)$

   $y + 7 = -\frac{2}{3}x - 4$

   This step utilizes the distributive property of multiplication over addition, a fundamental concept in algebra. By carefully multiplying the slope with each term inside the parentheses, we begin to unravel the equation and move closer to isolating y.

2. Isolate y by subtracting 7 from both sides of the equation:

   $y + 7 - 7 = -\frac{2}{3}x - 4 - 7$

   $y = -\frac{2}{3}x - 11$

   This step employs the principle of equality, which dictates that performing the same operation on both sides of an equation maintains its balance. By subtracting 7, we effectively eliminate the constant term on the left side, leaving y isolated and revealing the equation in its slope-intercept form.

Now the equation is in slope-intercept form, $y = -\frac{2}{3}x - 11$. We can see that the slope is $-\frac{2}{3}$ (which we already knew from the point-slope form) and the y-intercept is -11. This transformation allows us to clearly identify the key characteristics of the line: its steepness (slope) and where it intersects the y-axis (y-intercept).

Expressing the Linear Function

Once we have the equation in slope-intercept form, $y = -\frac{2}{3}x - 11$, we can express it as a linear function using function notation. Function notation is a way of representing equations that highlights the relationship between the input (x) and the output (y). We replace y with $f(x)$, which reads as "f of x." This notation emphasizes that the value of y depends on the value of x.

So, the linear function that represents the given line is:

$f(x) = -\frac{2}{3}x - 11$

This form is particularly useful for evaluating the function at different values of x. For example, to find the value of the function when $x = 0$, we simply substitute 0 for x:

$f(0) = -\frac{2}{3}(0) - 11 = -11$

This confirms that the y-intercept is indeed -11, as we identified earlier from the slope-intercept form. Expressing the linear equation as a function allows us to seamlessly integrate it into various mathematical contexts, such as graphing, analysis, and modeling real-world phenomena.

Identifying the Correct Option

Now that we have derived the linear function $f(x) = -\frac{2}{3}x - 11$, we can compare it to the given options and identify the correct one.

Looking at the options:

A. $f(x) = -\frac{2}{3}x - 11$ B. $f(x) = \frac{2}{3}x - 1$ C. $f(x) = \frac{2}{3}x + 3$ D. $f(x) = \frac{2}{3}x + 13$

We can see that option A matches our derived function exactly. Therefore, the correct answer is:

A. $f(x) = -\frac{2}{3}x - 11$

This step is the culmination of our journey, where we confidently match our derived solution with the provided options. It reinforces the importance of careful calculation and attention to detail throughout the process.

Conclusion

In this exploration, we successfully transformed the point-slope equation $y + 7 = -\frac{2}{3}(x + 6)$ into the slope-intercept form and expressed it as a linear function, $f(x) = -\frac{2}{3}x - 11$. This process involved understanding the point-slope form, applying algebraic manipulations to isolate y, and expressing the result in function notation. By mastering these techniques, you can confidently convert between different forms of linear equations and gain a deeper understanding of their properties. The ability to manipulate and interpret linear equations is a fundamental skill in mathematics, with applications spanning various fields, from physics and engineering to economics and computer science. So, continue to practice and explore the fascinating world of linear functions!

This journey through the transformation of linear equations highlights the interconnectedness of mathematical concepts. The point-slope form, slope-intercept form, and function notation are all different ways of representing the same underlying relationship. By understanding these forms and how to convert between them, we gain a more comprehensive and flexible understanding of linear functions.