Standard Form Equation Of A Line Through Two Points

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In the realm of mathematics, determining the equation of a line is a fundamental skill. Whether you're navigating geometric landscapes or analyzing data trends, understanding how to represent a line algebraically is crucial. This article will walk you through the process of finding the standard form equation of a line, using the points (8, -1) and (2, -5) as an example, and given the slope-intercept form y + 1 = (2/3)(x - 8).

1. Understanding the Basics: Slope-Intercept Form and Standard Form

Before diving into the calculations, let's solidify our understanding of the two key forms of linear equations we'll be working with:

  • Slope-Intercept Form: This form, expressed as y = mx + b, readily reveals the slope (m) and y-intercept (b) of the line. The slope (m) signifies the steepness and direction of the line, while the y-intercept (b) indicates the point where the line crosses the vertical y-axis. The given equation, y + 1 = (2/3)(x - 8), is closely related to this form and provides us with valuable information about the line's characteristics. This form is intuitive for visualizing and understanding the line's behavior on a graph. By simply observing the coefficients, we can immediately grasp the line's direction (positive or negative slope) and where it intersects the y-axis. This makes slope-intercept form a powerful tool for quickly analyzing and comparing different lines.

  • Standard Form: The standard form equation of a line is represented as Ax + By = C, where A, B, and C are integers, and A is typically a positive integer. This form is particularly useful for various algebraic manipulations and solving systems of linear equations. While it doesn't explicitly display the slope and y-intercept, standard form offers advantages in certain contexts. For instance, it simplifies the process of finding x and y-intercepts, which are crucial for graphing the line. Setting y = 0 allows us to solve for the x-intercept, and setting x = 0 allows us to solve for the y-intercept. Moreover, standard form is often preferred when dealing with systems of equations, as it facilitates the use of methods like elimination to find solutions. The goal of this article is to transform the given equation into this standard form, ensuring that A, B, and C are integers and A is positive.

2. Extracting Information from the Slope-Intercept Form

We are given the equation y + 1 = (2/3)(x - 8), which is a variation of the slope-intercept form. While not directly in the y = mx + b format, it provides us with essential information.

  • The Slope: By observing the coefficient of (x - 8), we can identify the slope, m, as 2/3. This indicates that for every 3 units the line moves horizontally, it rises 2 units vertically. The positive slope signifies that the line is increasing as we move from left to right on the graph. Understanding the slope is crucial for visualizing the line's direction and steepness. A larger slope value implies a steeper line, while a smaller slope value indicates a gentler incline. The slope is a fundamental property of a line that dictates its orientation in the coordinate plane.

  • A Point on the Line: The equation also reveals a point that the line passes through. The form y + 1 = (2/3)(x - 8) can be seen as a point-slope form, where the point is (8, -1). This means that when x is 8, y is -1. This point acts as an anchor for the line, fixing its position in the coordinate plane. Knowing a point on the line, along with the slope, is sufficient to completely define the line's path. This concept is fundamental to understanding the geometry of lines and their representations in different forms.

3. Converting to Slope-Intercept Form (y = mx + b)

To make the conversion to standard form easier, let's first transform the given equation into the familiar slope-intercept form (y = mx + b). This involves isolating y on one side of the equation.

  1. Distribute the slope: Multiply (2/3) by both x and -8 within the parentheses: y + 1 = (2/3)x - (2/3) * 8 y + 1 = (2/3)x - 16/3

    This step expands the equation, separating the x term and the constant term. Distributing the slope allows us to clearly see the individual components that contribute to the line's position and orientation. The result is a more explicit representation of the linear relationship between x and y.

  2. Isolate y: Subtract 1 from both sides of the equation: y = (2/3)x - 16/3 - 1

    Subtracting 1 from both sides effectively isolates y, bringing us closer to the slope-intercept form. This step maintains the equation's balance while reorganizing the terms to highlight the relationship between y and x. The goal is to express y as a function of x, which is precisely what the slope-intercept form achieves.

  3. Simplify the constant term: Convert 1 to 3/3 and combine it with -16/3: y = (2/3)x - 16/3 - 3/3 y = (2/3)x - 19/3

    Combining the constant terms simplifies the equation and expresses the y-intercept as a single fraction. This step consolidates the numerical values, making the equation more concise and easier to work with in subsequent steps. The simplified constant term represents the point where the line intersects the y-axis, which is a crucial piece of information for graphing and analyzing the line.

Now we have the equation in slope-intercept form: y = (2/3)x - 19/3. This form confirms our earlier observation about the slope (2/3) and explicitly shows the y-intercept (-19/3).

4. Converting to Standard Form (Ax + By = C)

With the equation in slope-intercept form, we can now convert it to standard form (Ax + By = C). Remember that in standard form, A, B, and C must be integers, and A should be positive.

  1. Eliminate the fraction: Multiply both sides of the equation by 3 to eliminate the denominator: 3 * y = 3 * [(2/3)x - 19/3] 3y = 2x - 19

    Multiplying by the denominator is a crucial step in eliminating fractions and ensuring that the coefficients in the standard form are integers. This step transforms the equation into a more manageable form, removing the complexity of fractional coefficients. The resulting equation maintains the same linear relationship but is expressed in whole numbers, which is a requirement for standard form.

  2. Rearrange the terms: Subtract 2x from both sides to move the x term to the left side: -2x + 3y = -19

    Rearranging the terms is a key step in aligning the equation with the Ax + By = C format. Moving the x term to the left side and the constant term to the right side sets the stage for the final transformation into standard form. This step involves algebraic manipulation to group like terms together, facilitating the identification of the coefficients A, B, and C.

  3. Make A positive: Multiply both sides of the equation by -1 to make the coefficient of x positive: (-1) * (-2x + 3y) = (-1) * (-19) 2x - 3y = 19

    Multiplying by -1 ensures that the coefficient of x, represented by A, is positive, which is a convention in standard form. This step completes the transformation into the standard form equation, satisfying the requirement that A be a positive integer. The resulting equation is a concise and standardized representation of the line, making it easier to compare with other linear equations and use in further calculations.

Therefore, the equation of the line in standard form is 2x - 3y = 19.

5. Verification and Conclusion

To ensure our answer is correct, we can substitute the original points (8, -1) and (2, -5) into the standard form equation 2x - 3y = 19 and verify that they satisfy the equation.

  • For (8, -1): 2(8) - 3(-1) = 16 + 3 = 19 (Correct)
  • For (2, -5): 2(2) - 3(-5) = 4 + 15 = 19 (Correct)

Both points satisfy the equation, confirming that our solution is accurate.

In conclusion, we successfully converted the given equation from a variation of slope-intercept form to standard form. The process involved distributing the slope, isolating y, eliminating fractions, rearranging terms, and ensuring that the coefficient of x was positive. This step-by-step guide provides a clear understanding of how to find the standard form equation of a line, a fundamental skill in mathematics and various applications.

By mastering this process, you gain a deeper understanding of linear equations and their different representations, empowering you to solve a wider range of mathematical problems and real-world scenarios.