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

Understanding Linear Functions and Point-Slope Form

To determine which linear function represents the line given by the point-slope equation y + 7 = (2/3)(x + 6), it's crucial to first understand the fundamental concepts of linear functions and the point-slope form of a linear equation. This knowledge will serve as the foundation for accurately transforming the given equation into slope-intercept form, which directly corresponds to a linear function. A linear function is a function whose graph is a straight line. It can be expressed in several forms, the most common being slope-intercept form, which is y = mx + b, where m represents the slope of the line and b represents the y-intercept. The slope indicates the steepness and direction of the line, while the y-intercept is the point where the line crosses the vertical y-axis. Understanding these parameters is key to analyzing and comparing linear equations.

The point-slope form is another way to represent a linear equation, particularly useful when you know a point on the line and the slope. This form is expressed as y - y1 = m(x - x1), where (x1, y1) is a specific point on the line and m is the slope. The point-slope form directly incorporates a known point and the slope, making it convenient for constructing the equation of a line when these pieces of information are readily available. Converting from point-slope form to slope-intercept form involves algebraic manipulation to isolate y on one side of the equation. This process typically involves distributing the slope, combining like terms, and adding or subtracting constants to both sides of the equation. This conversion is essential for expressing the linear equation in a form that directly reveals the slope and y-intercept, allowing for easy comparison with other linear functions and graphical representation.

The connection between these forms is vital for solving problems like the one presented. The point-slope form provides a direct way to write the equation of a line given a point and the slope, while the slope-intercept form allows for easy identification of the slope and y-intercept, which are key characteristics of a linear function. The ability to convert between these forms allows us to analyze and compare linear equations effectively. Therefore, by mastering these concepts and the conversion process, we can confidently address the problem at hand and identify the correct linear function.

Transforming the Point-Slope Equation to Slope-Intercept Form

The core of solving this problem lies in transforming the given point-slope equation, y + 7 = (2/3)(x + 6), into the slope-intercept form, y = mx + b. This transformation involves a series of algebraic steps, each meticulously executed to preserve the equation's integrity while revealing its underlying linear function. The initial step in this transformation is to distribute the slope, which is 2/3, across the terms within the parentheses on the right side of the equation. This means multiplying 2/3 by both x and 6. Performing this multiplication yields the following equation: *y + 7 = (2/3)x + (2/3)6. Simplifying the second term on the right side, *(2/3)6, gives us 4. Therefore, the equation now becomes y + 7 = (2/3)x + 4.

Next, our objective is to isolate y on the left side of the equation to achieve the slope-intercept form. To do this, we need to eliminate the +7 term on the left side. We accomplish this by subtracting 7 from both sides of the equation. This maintains the balance of the equation while moving us closer to the desired form. Subtracting 7 from both sides gives us: y + 7 - 7 = (2/3)x + 4 - 7. Simplifying this results in: y = (2/3)x - 3. This equation is now in slope-intercept form, where we can clearly identify the slope as 2/3 and the y-intercept as -3.

This transformation process highlights the importance of careful algebraic manipulation. Each step, from distributing the slope to subtracting constants, must be performed accurately to ensure the resulting equation is equivalent to the original. By systematically applying these steps, we have successfully converted the point-slope equation into slope-intercept form, making it easy to identify the corresponding linear function. This ability to manipulate equations is a fundamental skill in algebra and is crucial for solving a wide range of mathematical problems. The resulting equation, y = (2/3)x - 3, directly corresponds to a linear function, which we can now compare to the answer choices provided.

Identifying the Correct Linear Function

Now that we have transformed the point-slope equation y + 7 = (2/3)(x + 6) into the slope-intercept form y = (2/3)x - 3, the next step is to identify the correct linear function from the given options. The slope-intercept form, y = mx + b, directly corresponds to a linear function written in function notation as f(x) = mx + b. This means that the equation we derived, y = (2/3)x - 3, can be directly translated into a linear function by replacing y with f(x). This gives us the linear function f(x) = (2/3)x - 3.

To find the correct answer, we need to compare this derived linear function, f(x) = (2/3)x - 3, with the options provided. This involves carefully examining each option to see which one matches our result. The options are:

A. f(x) = (2/3)x - 11 B. f(x) = (2/3)x - 1 C. f(x) = (2/3)x + 3 D. f(x) = (2/3)x + 13

By direct comparison, we can see that none of the provided options exactly match our derived function, f(x) = (2/3)x - 3. This indicates a potential error in the provided options or a mistake in our calculations. Let's re-examine the transformation process to ensure accuracy. We started with y + 7 = (2/3)(x + 6), distributed the 2/3 to get y + 7 = (2/3)x + 4, and then subtracted 7 from both sides to get y = (2/3)x - 3. Our calculations appear to be correct.

Given this discrepancy, it's essential to consider the possibility of an error in the provided answer choices. If we are confident in our calculations, we must conclude that none of the options accurately represent the linear function derived from the given point-slope equation. This underscores the importance of careful attention to detail in both the transformation process and the comparison step. While we cannot select an answer from the given options, we have successfully determined the correct linear function based on the initial equation. This highlights the importance of understanding the underlying mathematical principles and applying them accurately, even when faced with potentially incorrect options.

Conclusion and Final Answer

In conclusion, to determine the linear function represented by the point-slope equation y + 7 = (2/3)(x + 6), we systematically transformed the equation into slope-intercept form, y = mx + b. This involved distributing the slope, 2/3, and isolating y to obtain the equation y = (2/3)x - 3. This equation directly translates to the linear function f(x) = (2/3)x - 3. Comparing this derived function with the provided options revealed that none of them exactly matched our result. This discrepancy indicates a potential error in the answer choices.

Despite the absence of a matching option, the process of converting the point-slope equation to slope-intercept form and then expressing it as a linear function is a valuable exercise in algebraic manipulation and understanding the relationship between different forms of linear equations. This ability to transform equations and accurately identify the corresponding linear function is a fundamental skill in mathematics. It is crucial for solving a variety of problems, from simple equation solving to more complex mathematical modeling.

Therefore, while we cannot select an answer from the provided options, we have confidently determined that the linear function represented by the given point-slope equation is f(x) = (2/3)x - 3. This result underscores the importance of accuracy in mathematical calculations and the ability to identify and address discrepancies when they arise. It also highlights the value of understanding the underlying principles and applying them correctly, even when faced with potentially flawed information. The final answer, based on our calculations, is f(x) = (2/3)x - 3, although this option is not available among the choices provided.