Rewriting 2x - 10y = -20 Into Slope-Intercept Form A Step-by-Step Guide

Understanding the slope-intercept form is a fundamental skill in algebra, allowing us to quickly identify a line's slope and y-intercept. The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope and b represents the y-intercept. Mastering this form provides a clear and concise way to visualize and analyze linear relationships. This article will delve into the process of rewriting the equation 2x - 10y = -20 into slope-intercept form, providing a detailed, step-by-step explanation to ensure clarity and comprehension. We will explore the underlying algebraic principles and demonstrate how to isolate y to achieve the desired form. The importance of this skill extends beyond the classroom, playing a vital role in various fields such as physics, engineering, economics, and computer science, where linear models are frequently used to represent and solve real-world problems. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide will equip you with the knowledge and confidence to tackle linear equations effectively.

Step 1: Isolate the 'y' Term

Our initial equation, 2x - 10y = -20, presents a linear relationship between x and y. To convert this equation into slope-intercept form, the first crucial step involves isolating the term containing y. This means we need to strategically manipulate the equation to get the y term by itself on one side. The standard approach is to subtract the x term from both sides of the equation, ensuring we maintain the balance and equality of the expression. In our case, we subtract 2x from both sides of the equation, which yields: -10y = -2x - 20. This step is critical because it begins the process of unraveling the equation to reveal the slope and y-intercept in a clear and interpretable manner. The principle behind this step is rooted in the fundamental properties of equality, which dictate that performing the same operation on both sides of an equation preserves its validity. Isolating the y term is not just a mechanical procedure; it's a strategic move that sets the stage for the subsequent steps, ultimately leading us to the desired slope-intercept form. This process lays the groundwork for easily identifying the line's key characteristics, making it a cornerstone of linear equation manipulation.

Step 2: Divide to Solve for 'y'

After isolating the term with y in the previous step, we now have the equation -10y = -2x - 20. To completely isolate y, we must eliminate the coefficient that multiplies it. In this case, the coefficient is -10. The algebraic operation to undo multiplication is division. Therefore, we divide both sides of the equation by -10. This step is crucial as it solves for y, expressing it in terms of x and a constant. When dividing, we must ensure that every term on both sides of the equation is divided by -10. This maintains the equality and ensures the resulting equation is equivalent to the original. Performing this division, we get: y = (-2x / -10) + (-20 / -10). This fraction simplifies to y = (1/5)x + 2. The division step is not just about isolating y; it's about transforming the equation into a format where the slope and y-intercept are readily visible. By dividing by the coefficient of y, we unveil the relationship between y and x in its simplest form, paving the way for easy interpretation and application of the linear equation.

Step 3: Simplify the Equation

Following the division, our equation now reads y = (-2x / -10) + (-20 / -10). The next crucial step is to simplify the fractions within the equation. Simplification involves reducing fractions to their lowest terms and performing any necessary arithmetic operations. In this case, we have two fractions to simplify: -2x / -10 and -20 / -10. Both fractions involve negative numbers, and a negative divided by a negative results in a positive. The fraction -2x / -10 simplifies to (1/5)x, and the fraction -20 / -10 simplifies to 2. Therefore, the simplified equation becomes y = (1/5)x + 2. This simplification step is not merely cosmetic; it's essential for clarity and ease of interpretation. By reducing the fractions, we present the equation in its most straightforward form, making it easier to identify the slope and y-intercept. Simplification enhances understanding and reduces the likelihood of errors in subsequent calculations or applications of the equation. A simplified equation is more accessible and reveals the underlying mathematical relationships more clearly, which is a cornerstone of effective mathematical communication and problem-solving.

Step 4: Identify the Slope and Y-intercept

After simplifying the equation to y = (1/5)x + 2, we have successfully transformed it into slope-intercept form, which is y = mx + b. In this standard form, m represents the slope of the line, and b represents the y-intercept. The slope, m, indicates the rate of change of y with respect to x, essentially describing the steepness and direction of the line. The y-intercept, b, is the point where the line crosses the y-axis, occurring when x is equal to 0. By comparing our simplified equation y = (1/5)x + 2 with the general form y = mx + b, we can easily identify that the slope, m, is 1/5, and the y-intercept, b, is 2. This means that for every 5 units we move along the x-axis, the line rises 1 unit on the y-axis. Additionally, the line intersects the y-axis at the point (0, 2). Identifying the slope and y-intercept is the primary benefit of converting an equation to slope-intercept form. These two values provide a wealth of information about the line's behavior and position on the coordinate plane, making it easier to graph, analyze, and compare with other lines. This step is crucial for understanding the linear relationship and its implications.

Conclusion: The Power of Slope-Intercept Form

In conclusion, we have successfully rewritten the equation 2x - 10y = -20 into slope-intercept form, which is y = (1/5)x + 2. This process involved isolating the y term, dividing to solve for y, and simplifying the resulting equation. Through these steps, we were able to identify the slope as 1/5 and the y-intercept as 2. The slope-intercept form, y = mx + b, is a powerful tool in algebra for several reasons. First, it provides a clear and immediate understanding of the line's characteristics: the slope (m) indicates its steepness and direction, and the y-intercept (b) indicates where the line crosses the y-axis. This form makes it easy to graph the line, as we only need two points – the y-intercept and another point derived from the slope – to draw a straight line. Furthermore, the slope-intercept form simplifies the comparison of different linear equations. By looking at the slopes and y-intercepts, we can quickly determine if lines are parallel (same slope), perpendicular (slopes are negative reciprocals), or intersecting. The ability to rewrite equations into slope-intercept form is a fundamental skill that has wide-ranging applications in mathematics and various real-world contexts. From physics and engineering to economics and computer science, linear equations and their graphical representations are used to model and solve a vast array of problems. Mastering this skill empowers students and professionals alike to analyze and interpret linear relationships effectively.