Transforming Linear Equations How To Convert 2x - 5y + 15 = 0 To Slope-Intercept Form
In the realm of linear equations, the slope-intercept form stands as a cornerstone, providing a clear and concise representation of a line's characteristics. This form, expressed as y = mx + b, elegantly reveals both the slope (m) and the y-intercept (b) of the line, making it a powerful tool for analysis and graphing. Our exploration today delves into the transformation of a linear equation from its standard form into the coveted slope-intercept form. Specifically, we will tackle the equation 2x - 5y + 15 = 0, dissecting the steps required to rewrite it in the form y = mx + b. This process not only solidifies our understanding of linear equations but also highlights the algebraic manipulations that underpin mathematical problem-solving. Mastering this conversion is essential for anyone seeking to grasp the intricacies of linear relationships and their graphical representations. This detailed guide will walk you through each step, ensuring you gain a comprehensive understanding of how to convert a standard form equation to slope-intercept form.
The slope-intercept form, y = mx + b, is a fundamental concept in linear algebra. In this equation, m represents the slope of the line, which indicates its steepness and direction, while b represents the y-intercept, the point where the line crosses the y-axis. The beauty of this form lies in its simplicity and the direct information it provides about the line's behavior. The slope, often described as "rise over run," quantifies how much the y-value changes for every unit change in the x-value. A positive slope indicates an upward slant, while a negative slope indicates a downward slant. The magnitude of the slope reflects the steepness of the line; a larger absolute value signifies a steeper line. The y-intercept, on the other hand, is a specific point (0, b) on the coordinate plane, marking where the line intersects the vertical axis. Knowing the slope and y-intercept allows us to easily graph the line or visualize its path. To fully appreciate the power of the slope-intercept form, let's consider its advantages. First, it makes graphing lines incredibly straightforward. By plotting the y-intercept and using the slope to find additional points, we can quickly sketch the line. Second, it allows for easy comparison of different lines. By simply comparing their slopes and y-intercepts, we can determine if lines are parallel, perpendicular, or intersecting. Third, it is a useful form for modeling real-world situations. Many relationships in science, engineering, and economics can be approximated by linear equations, and the slope-intercept form provides a natural way to interpret these relationships. For instance, the equation might represent the cost of a service based on a fixed fee (y-intercept) and a per-unit charge (slope).
Our primary goal is to transform the given equation, 2x - 5y + 15 = 0, into the y = mx + b format. This involves strategically isolating the 'y' variable on one side of the equation. The process begins by addressing the terms that do not involve 'y'. In our case, these are the 2x and +15 terms. To move these terms to the other side of the equation, we employ the fundamental principle of algebraic manipulation: performing the same operation on both sides to maintain equality. Therefore, we subtract 2x and 15 from both sides of the equation. This yields a new equation: -5y = -2x - 15. Notice how the signs of the terms change as they cross the equals sign. The positive 2x becomes negative, and the positive 15 also becomes negative. This is a crucial step in isolating 'y'. Next, we need to eliminate the coefficient multiplying 'y', which is -5. To achieve this, we divide both sides of the equation by -5. This step is essential to get 'y' by itself. When dividing, we must remember to distribute the division to each term on the right-hand side. This means we divide both -2x and -15 by -5. Performing this division gives us: y = (-2x / -5) + (-15 / -5). Simplifying the fractions, we get y = (2/5)x + 3. This is now in the desired slope-intercept form, y = mx + b, where the slope m is 2/5 and the y-intercept b is 3. This step-by-step isolation of 'y' demonstrates the power of algebraic manipulation in solving equations. By carefully applying inverse operations, we can transform equations into more useful forms, revealing key information about the relationships they represent.
Now that we have successfully transformed the equation 2x - 5y + 15 = 0 into its slope-intercept form, y = (2/5)x + 3, we can readily identify the key characteristics of the line: its slope and y-intercept. Recall that the slope-intercept form is y = mx + b, where m represents the slope and b represents the y-intercept. By direct comparison, we can see that the coefficient of x in our transformed equation is 2/5. Therefore, the slope of the line is 2/5. This positive slope indicates that the line rises from left to right. For every 5 units we move horizontally, the line moves 2 units vertically. This ratio gives us a precise measure of the line's steepness. Next, we identify the y-intercept. In the equation y = (2/5)x + 3, the constant term is 3. This means that the y-intercept is the point (0, 3). This is the point where the line crosses the y-axis. Knowing the slope and y-intercept provides a complete picture of the line's position and orientation in the coordinate plane. With this information, we can easily graph the line by first plotting the y-intercept and then using the slope to find another point. For instance, starting at (0, 3), we can move 5 units to the right and 2 units up to find another point on the line, (5, 5). Connecting these two points gives us the line represented by the equation 2x - 5y + 15 = 0. Identifying the slope and y-intercept is not only useful for graphing but also for understanding the behavior of the linear relationship represented by the equation. The slope tells us how the dependent variable (y) changes in response to changes in the independent variable (x), while the y-intercept gives us the value of the dependent variable when the independent variable is zero. These concepts are fundamental in many applications of linear equations in science, engineering, and economics.
Having derived the slope-intercept form of the equation 2x - 5y + 15 = 0 as y = (2/5)x + 3, we now turn our attention to the provided options to determine the correct answer. The options presented are:
- A. y = (2/5)x - 3
- B. y = (2/5)x + 3
- C. y = (-2/5)x + 3
By carefully comparing our derived equation with the options, we can see a clear match with option B. Both equations have the same slope, 2/5, and the same y-intercept, 3. This confirms that y = (2/5)x + 3 is the correct slope-intercept form of the given equation. Options A and C, on the other hand, differ in either the y-intercept or the slope. Option A has a y-intercept of -3, while option C has a slope of -2/5. These differences highlight the importance of accurate algebraic manipulation when transforming equations. A small error in sign or division can lead to an incorrect result. The process of comparing our solution with the given options serves as a crucial verification step. It ensures that we have not made any mistakes in our calculations and that we have arrived at the correct answer. In multiple-choice questions, this step is particularly important as it allows us to eliminate incorrect options and build confidence in our final choice. Moreover, this comparison reinforces our understanding of the slope-intercept form. By analyzing why options A and C are incorrect, we gain a deeper appreciation for the significance of the slope and y-intercept in defining a line.
In this comprehensive exploration, we successfully transformed the linear equation 2x - 5y + 15 = 0 into its slope-intercept form, y = (2/5)x + 3. This transformation involved a series of algebraic manipulations, including isolating the 'y' variable by subtracting terms and dividing by the coefficient of 'y'. We then identified the slope as 2/5 and the y-intercept as 3, which provided us with a clear understanding of the line's characteristics. By comparing our result with the given options, we confidently selected the correct answer: y = (2/5)x + 3. The ability to convert equations into slope-intercept form is a fundamental skill in algebra and is essential for understanding and graphing linear relationships. The slope-intercept form not only simplifies the process of graphing lines but also provides valuable insights into the behavior of the linear relationship. The slope tells us how the dependent variable changes with respect to the independent variable, while the y-intercept gives us the starting value of the dependent variable. Mastering this skill opens doors to a wide range of applications in mathematics, science, engineering, and economics. From modeling real-world phenomena to solving complex problems, the ability to work with linear equations in slope-intercept form is a powerful tool. This exercise has not only reinforced our understanding of algebraic manipulation but has also highlighted the importance of accuracy and attention to detail in mathematical problem-solving. By following a systematic approach and verifying our results, we can confidently tackle similar challenges in the future.
