Converting Point-Slope Form To Slope-Intercept Form Explained

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When dealing with linear equations, the point-slope form emerges as a powerful tool for representing a line. It elegantly captures the relationship between a line's slope and a specific point it traverses. The general form of the point-slope equation is expressed as:

y - y₁ = m(x - x₁)

where:

  • 'm' denotes the slope of the line, a measure of its steepness and direction.
  • '(x₁, y₁)' represents a known point that lies on the line.

This form proves particularly useful when we are given a point on the line and its slope, or when we have two points through which the line passes. In this article, we have the equation of the line in point-slope form: y - 3 = (1/2)(x - 1). Our mission is to transform this equation into the slope-intercept form, which offers a different perspective on the line's characteristics.

The Allure of Slope-Intercept Form

The slope-intercept form of a linear equation presents itself as another valuable way to express the relationship between variables. It takes the form:

y = mx + b

where:

  • 'm' retains its role as the slope of the line.
  • 'b' represents the y-intercept, the point where the line intersects the y-axis. This form provides a direct visualization of the line's slope and its vertical position on the coordinate plane. Transforming an equation into slope-intercept form often simplifies graphing and analyzing linear relationships. Understanding the slope-intercept form helps us quickly grasp the line's behavior and its position in the coordinate plane. It allows us to easily identify the slope and y-intercept, which are crucial parameters for understanding the line's characteristics. The slope tells us how steeply the line rises or falls, while the y-intercept indicates where the line crosses the vertical axis. This form is particularly useful for comparing different lines and understanding their relative positions and orientations. The slope-intercept form makes it straightforward to graph the line. By plotting the y-intercept and using the slope to find another point, we can easily draw the line. This visual representation is invaluable for understanding the line's behavior and its relationship to other lines or points. Moreover, the slope-intercept form is widely used in various applications of linear equations, including modeling real-world phenomena, solving systems of equations, and performing linear regression analysis. Its versatility and ease of interpretation make it a fundamental concept in mathematics and related fields.

The journey from point-slope form to slope-intercept form involves a series of algebraic manipulations. These steps systematically isolate 'y' on one side of the equation, revealing the slope and y-intercept in the desired format. The given equation in point-slope form is y - 3 = (1/2)(x - 1). Our goal is to rewrite this equation in the form y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. Let's embark on this transformation step by step. First, we distribute the (1/2) on the right side of the equation: y - 3 = (1/2)x - (1/2). This step expands the expression and prepares us to isolate 'y'. Next, we add 3 to both sides of the equation to isolate 'y' on the left side: y - 3 + 3 = (1/2)x - (1/2) + 3. This operation maintains the equality of the equation while moving the constant term to the right side. Now, we simplify the equation: y = (1/2)x - (1/2) + 3. To combine the constant terms, we need a common denominator. We rewrite 3 as 6/2: y = (1/2)x - (1/2) + (6/2). Finally, we combine the constant terms: y = (1/2)x + (5/2). This is the slope-intercept form of the equation. From this form, we can directly identify the slope as 1/2 and the y-intercept as 5/2. Therefore, the line has a slope of 1/2 and intersects the y-axis at the point (0, 5/2). This transformation process highlights the power of algebraic manipulation in revealing different aspects of a linear equation. By converting from point-slope form to slope-intercept form, we gain a clearer understanding of the line's slope and y-intercept, which are essential for graphing and analyzing linear relationships. The slope-intercept form provides a direct visualization of the line's characteristics, making it a valuable tool in various mathematical and real-world applications.

Step-by-Step Conversion

Let's delve into the transformation process with meticulous detail:

  1. Distribute: Begin by distributing the slope (1/2) on the right side of the equation: y - 3 = (1/2)x - (1/2) This step expands the equation, preparing it for further manipulation.
  2. Isolate 'y': To isolate 'y', add 3 to both sides of the equation: y - 3 + 3 = (1/2)x - (1/2) + 3 This action moves the constant term to the right side, bringing us closer to the slope-intercept form.
  3. Simplify: Simplify the equation by combining the constant terms. To do this, express 3 as a fraction with a denominator of 2: y = (1/2)x - (1/2) + (6/2)
  4. Combine Constants: Combine the constant terms: y = (1/2)x + (5/2) This final step yields the equation in slope-intercept form.

The Grand Reveal: Slope and Y-Intercept

In the slope-intercept form, y = (1/2)x + (5/2), the coefficients unveil crucial information about the line:

  • Slope (m): The coefficient of 'x', which is 1/2, represents the slope of the line. This indicates that for every 2 units the line moves horizontally, it rises 1 unit vertically.
  • Y-intercept (b): The constant term, 5/2, signifies the y-intercept. This is the point where the line intersects the y-axis, specifically at the coordinates (0, 5/2).

By meticulously transforming the point-slope form into slope-intercept form, we arrive at the equation y = (1/2)x + (5/2). Comparing this result with the provided options, we can confidently identify the correct answer.

Examining the Options

Let's revisit the options presented:

  • A. y = (1/2)x + 2
  • B. y = (1/2)x - 4
  • C. y = (1/2)x + (5/2)

Our derived equation, y = (1/2)x + (5/2), perfectly aligns with option C. Therefore, option C is the correct slope-intercept form of the given equation.

The journey from point-slope form to slope-intercept form demonstrates the power of algebraic manipulation in revealing different perspectives of a linear equation. By systematically isolating 'y', we unveiled the slope and y-intercept, key characteristics that define the line's behavior and position. This process not only reinforces our understanding of linear equations but also equips us with a valuable tool for analyzing and interpreting mathematical relationships. Transforming the equation y - 3 = (1/2)(x - 1) from point-slope form to slope-intercept form involved a series of algebraic steps. We began by distributing the slope (1/2) on the right side of the equation, which expanded the expression and prepared it for further manipulation. Next, we isolated 'y' by adding 3 to both sides of the equation, moving the constant term to the right side. This step brought us closer to the desired slope-intercept form. Then, we simplified the equation by combining the constant terms. To do this, we expressed 3 as a fraction with a denominator of 2, allowing us to add it to the other constant term. Finally, we combined the constant terms, resulting in the equation y = (1/2)x + (5/2). This equation is in slope-intercept form, where the coefficient of 'x' (1/2) represents the slope of the line and the constant term (5/2) represents the y-intercept. By comparing this result with the provided options, we confidently identified the correct answer as option C, y = (1/2)x + (5/2). This exercise highlights the importance of algebraic manipulation in revealing different aspects of a linear equation. The slope-intercept form provides a direct visualization of the line's slope and y-intercept, making it a valuable tool for graphing and analyzing linear relationships. Understanding the transformation process from point-slope form to slope-intercept form enhances our ability to work with linear equations in various mathematical and real-world applications. The point-slope form is particularly useful when we know a point on the line and its slope, or when we have two points through which the line passes. However, the slope-intercept form is often preferred for its simplicity and ease of interpretation. It allows us to quickly identify the slope and y-intercept, which are crucial parameters for understanding the line's characteristics. The ability to transform between these forms is a valuable skill in mathematics, enabling us to choose the most convenient representation for a given problem or situation. In conclusion, the transformation from point-slope form to slope-intercept form is a fundamental technique in linear algebra. It allows us to express a linear equation in a different format, revealing key information about the line's slope and y-intercept. This process enhances our understanding of linear relationships and provides a valuable tool for solving mathematical problems and modeling real-world phenomena.