Identify The Equation Not Representing The Line Through (3,-6) And (1,2)

Introduction

In the realm of coordinate geometry, lines are fundamental geometric objects, and their equations provide a powerful tool for describing their properties and behavior. When given two points in a plane, there exists a unique line that passes through them. Finding the equation of this line is a common task in mathematics, and there are several ways to represent the same line using different forms of equations. In this article, we will delve into the process of determining the equation of a line passing through two given points and explore how different equation forms can represent the same line. We will use the specific example of the line passing through the points $(3, -6)$ and $(1, 2)$ to illustrate these concepts and then identify which of the given options is not an equation of this line.

Finding the Slope of the Line

The slope of a line is a crucial parameter that indicates its steepness and direction. Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, the slope, often denoted by $m$, can be calculated using the formula:

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

In our case, the two points are $(3, -6)$ and $(1, 2)$. Let's designate $(3, -6)$ as $(x_1, y_1)$ and $(1, 2)$ as $(x_2, y_2)$. Plugging these values into the slope formula, we get:

$$m = \frac{2 - (-6)}{1 - 3} = \frac{2 + 6}{-2} = \frac{8}{-2} = -4$$

Therefore, the slope of the line passing through the points $(3, -6)$ and $(1, 2)$ is $-4$. This negative slope indicates that the line is decreasing as we move from left to right on the coordinate plane.

Point-Slope Form of a Line Equation

The point-slope form is a versatile way to represent the equation of a line. It utilizes the slope of the line and the coordinates of a single point on the line. The general form of the point-slope equation is:

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

where $(x_1, y_1)$ is a point on the line and $m$ is the slope. This form is particularly useful when we know the slope and one point on the line, as is our case. We have already calculated the slope, $m = -4$, and we have two points to choose from: $(3, -6)$ and $(1, 2)$. Let's use the point $(3, -6)$ first. Plugging the values into the point-slope form, we get:

$$y - (-6) = -4(x - 3)$$

Simplifying, we have:

$$y + 6 = -4(x - 3)$$

This equation represents the line passing through $(3, -6)$ and $(1, 2)$ in point-slope form. Now, let's use the other point, $(1, 2)$, and the slope $m = -4$ in the point-slope form:

$$y - 2 = -4(x - 1)$$

This is another valid equation for the same line, also in point-slope form. Both equations, $y + 6 = -4(x - 3)$ and $y - 2 = -4(x - 1)$, are equivalent and represent the same line. They simply use different points on the line to express the relationship between $x$ and $y$.

Slope-Intercept Form of a Line Equation

Another common way to represent the equation of a line is the slope-intercept form, which is given by:

$$y = mx + b$$

where $m$ is the slope and $b$ is the y-intercept (the y-coordinate of the point where the line intersects the y-axis). To convert the point-slope form equations we derived earlier into slope-intercept form, we can simply expand and rearrange the terms. Let's start with the equation $y + 6 = -4(x - 3)$:

$$y + 6 = -4x + 12$$

Subtracting 6 from both sides, we get:

$$y = -4x + 12 - 6$$

$$y = -4x + 6$$

Now, let's convert the other point-slope form equation, $y - 2 = -4(x - 1)$, into slope-intercept form:

$$y - 2 = -4x + 4$$

Adding 2 to both sides, we get:

$$y = -4x + 4 + 2$$

$$y = -4x + 6$$

As we can see, both point-slope form equations, when converted to slope-intercept form, yield the same equation: $y = -4x + 6$. This confirms that both point-slope equations indeed represent the same line.

Analyzing the Given Options

Now that we have determined the slope-intercept form of the line passing through $(3, -6)$ and $(1, 2)$ to be $y = -4x + 6$, and we have also derived two equivalent point-slope forms, we can compare these equations to the given options and identify the one that does not represent the same line. The given options are:

A. $y - 2 = -4(x - 1)$ B. $y = -4x + 6$ C. $y + 6 = -4(x - 3)$ D. $y - 1 = -4(x - 2)$

We have already derived equations A ($y - 2 = -4(x - 1)$) and B ($y = -4x + 6$) and C ($y + 6 = -4(x - 3)$) and found them to be valid representations of the line passing through $(3, -6)$ and $(1, 2)$. Therefore, we can confidently say that options A, B, and C are equations of the line.

Let's examine option D: $y - 1 = -4(x - 2)$. To determine if this equation represents the same line, we can convert it to slope-intercept form:

$$y - 1 = -4x + 8$$

Adding 1 to both sides, we get:

$$y = -4x + 8 + 1$$

$$y = -4x + 9$$

Comparing this equation, $y = -4x + 9$, with the slope-intercept form we derived earlier, $y = -4x + 6$, we can see that they have the same slope (-4) but different y-intercepts (9 and 6, respectively). This means that the lines represented by these equations are parallel but distinct. Therefore, option D, $y - 1 = -4(x - 2)$, does not represent the same line passing through $(3, -6)$ and $(1, 2)$.

Conclusion

In this comprehensive exploration, we have successfully determined the equation of the line passing through the points $(3, -6)$ and $(1, 2)$ using both point-slope and slope-intercept forms. We calculated the slope to be $-4$ and derived the equation $y = -4x + 6$ in slope-intercept form. We also found two equivalent point-slope forms: $y + 6 = -4(x - 3)$ and $y - 2 = -4(x - 1)$. By comparing these equations with the given options, we identified that option D, $y - 1 = -4(x - 2)$, is not an equation of the same line. This exercise highlights the importance of understanding different forms of line equations and how they can be manipulated to determine if they represent the same line.

This problem showcases the fundamental concepts of coordinate geometry, including the calculation of slope, the point-slope form, and the slope-intercept form of a line equation. It also emphasizes the importance of algebraic manipulation in converting between different forms of equations and verifying their equivalence. Understanding these concepts is crucial for further studies in mathematics and related fields.

In conclusion, the equation that does not represent the line passing through $(3,-6)$ and $(1,2)$ is D. $y-1=-4(x-2)$. This detailed explanation not only provides the answer but also reinforces the underlying principles and techniques involved in solving such problems.