Point-Slope Form Equation How To Find It

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This article delves into the concept of point-slope form, a fundamental tool in linear algebra, and provides a step-by-step guide on how to identify and construct equations in this form. We will address the question: "Which equation shows the point-slope form of the line that passes through (3,2) and has a slope of 1/3?" by dissecting the point-slope formula, exploring its components, and applying it to the given problem. Furthermore, we will discuss the significance of point-slope form in various mathematical contexts and highlight its advantages over other linear equation forms.

Decoding the Point-Slope Form

The point-slope form of a linear equation is a powerful representation that allows us to define a line using a single point on the line and its slope. The general formula for point-slope form is:

y - y1 = m(x - x1)

Where:

  • (x1, y1) represents a specific point on the line.
  • m represents the slope of the line, indicating its steepness and direction.
  • (x, y) represents any general point on the line.

The beauty of this form lies in its intuitive nature. It directly incorporates the concept of slope as the rate of change and utilizes a known point to anchor the line in the coordinate plane. Understanding each component is crucial for effectively using the point-slope form.

Unpacking the Components

To truly grasp the point-slope form, let's break down each component individually:

  1. The Point (x1, y1): This is a fixed point that the line passes through. It provides a concrete location on the line, acting as a reference point for defining the line's position in the coordinate plane. When you are given a point and asked to write an equation in point-slope form, this is the first piece of information you will use.

  2. The Slope (m): The slope, often denoted by m, quantifies the line's steepness and direction. It represents the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. A positive slope indicates an increasing line (going upwards from left to right), while a negative slope indicates a decreasing line. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line.

  3. The Variables (x, y): These variables represent any general point on the line. They are not fixed values but rather represent the coordinates of any point that satisfies the equation. The relationship between x and y, as defined by the equation, dictates all the points that lie on the line.

Why Use Point-Slope Form?

The point-slope form offers several advantages over other forms of linear equations, such as slope-intercept form (y = mx + b) and standard form (Ax + By = C):

  • Direct Incorporation of Given Information: It directly utilizes the given point and slope, making it straightforward to construct the equation when this information is available.
  • Conceptual Clarity: It highlights the fundamental relationship between a point on a line and its slope, providing a clear visual interpretation of the line's characteristics.
  • Ease of Conversion: It can be easily converted to other forms, such as slope-intercept form, by simple algebraic manipulation.

Solving the Problem: Finding the Correct Equation

Now, let's apply our understanding of the point-slope form to solve the problem at hand: "Which equation shows the point-slope form of the line that passes through (3,2) and has a slope of 1/3?"

We are given:

  • A point: (3, 2), which means x1 = 3 and y1 = 2
  • A slope: m = 1/3

Using the point-slope form equation:

y - y1 = m(x - x1)

Substitute the given values:

y - 2 = (1/3)(x - 3)

Comparing this equation with the given options:

A. y + 2 = (1/3)(x + 3) B. y - 2 = (1/3)(x - 3) C. y + 3 = (1/3)(x + 2) D. y - 3 = (1/3)(x - 2)

We can clearly see that option B matches the equation we derived. Therefore, the correct answer is:

y - 2 = (1/3)(x - 3)

This equation represents the line that passes through the point (3, 2) and has a slope of 1/3.

Analyzing the Incorrect Options

To further solidify our understanding, let's analyze why the other options are incorrect:

  • Option A: y + 2 = (1/3)(x + 3) This option incorrectly uses +2 instead of -2 and +3 instead of -3, indicating an incorrect substitution of the point (3, 2) into the point-slope form.
  • Option C: y + 3 = (1/3)(x + 2) This option incorrectly swaps the x and y coordinates and uses addition instead of subtraction, demonstrating a misunderstanding of the point-slope form.
  • Option D: y - 3 = (1/3)(x - 2) This option also incorrectly swaps the x and y coordinates, indicating an incorrect application of the point-slope formula.

By understanding why these options are wrong, we reinforce our understanding of the correct application of the point-slope form.

Applications of Point-Slope Form in Mathematics

The point-slope form is not just a theoretical concept; it has numerous practical applications in various mathematical fields, including:

1. Finding the Equation of a Line Given a Point and Slope

This is the most direct application of the point-slope form. As demonstrated in the solved problem, if you have a point and the slope of a line, you can easily construct the equation of the line in point-slope form.

For example, suppose you are given a point (-1, 4) and a slope of -2. Using the point-slope form, you can write the equation as:

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

y - 4 = -2(x + 1)

This equation represents the line that passes through (-1, 4) and has a slope of -2.

2. Determining the Equation of a Line Given Two Points

If you are given two points on a line, you can first calculate the slope using the formula:

m = (y2 - y1) / (x2 - x1)

Once you have the slope, you can choose either of the two points and use the point-slope form to write the equation of the line.

For instance, if the points are (2, 3) and (4, 7), the slope is:

m = (7 - 3) / (4 - 2) = 4 / 2 = 2

Using the point (2, 3) and the slope m = 2, the point-slope form equation is:

y - 3 = 2(x - 2)

3. Converting to Slope-Intercept Form

The point-slope form can be easily converted to slope-intercept form (y = mx + b), which is useful for identifying the y-intercept of the line. To convert from point-slope form, simply distribute the slope and solve for y.

Consider the equation y - 2 = (1/3)(x - 3). To convert it to slope-intercept form:

  1. Distribute the slope: y - 2 = (1/3)x - 1
  2. Add 2 to both sides: y = (1/3)x + 1

Now the equation is in slope-intercept form, and we can see that the y-intercept is 1.

4. Analyzing Parallel and Perpendicular Lines

Point-slope form is helpful in analyzing parallel and perpendicular lines. Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other. By comparing the slopes in the point-slope form equations, we can determine if lines are parallel, perpendicular, or neither.

5. Linear Modeling and Real-World Applications

In many real-world scenarios, linear relationships can be modeled using the point-slope form. For example, if you know the initial value and the rate of change of a quantity, you can use the point-slope form to model the quantity's behavior over time. This is particularly useful in fields like physics, economics, and engineering.

Conclusion

The point-slope form is a versatile and powerful tool in linear algebra. Its ability to directly incorporate a point and slope makes it invaluable for constructing and analyzing linear equations. By understanding the components of the point-slope form and its applications, you can effectively solve a wide range of mathematical problems and gain a deeper appreciation for the beauty and utility of linear equations. In the context of the question we addressed, the correct equation representing the line passing through (3,2) with a slope of 1/3 is indeed y - 2 = (1/3)(x - 3). This exercise highlights the importance of carefully applying the point-slope formula and understanding its underlying principles.

By mastering the point-slope form, you unlock a fundamental building block for more advanced mathematical concepts and applications. Whether you are solving problems in algebra, calculus, or real-world modeling, the point-slope form provides a clear and concise way to represent and analyze linear relationships.