Equation Of A Line Through (0, -3) With Slope -1/2 A Step-by-Step Solution

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In the realm of linear equations, determining the equation of a line given specific parameters is a fundamental skill. This article will guide you through the process of finding the equation of a line that passes through the point (0, -3) and has a slope of -1/2. We will delve into the slope-intercept form, point-slope form, and general form of linear equations, illustrating how to apply these concepts to arrive at the solution. Understanding these concepts is crucial for various applications in mathematics, physics, engineering, and other fields where linear relationships are prevalent.

Understanding the Slope-Intercept Form

One of the most common and intuitive ways to represent a linear equation is the slope-intercept form. This form is expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis). The slope m quantifies the steepness of the line and its direction (positive or negative), while the y-intercept b provides a specific point on the line. The beauty of the slope-intercept form lies in its simplicity and directness. By knowing the slope and y-intercept, you can immediately write the equation of the line. In our specific problem, we are given that the line passes through the point (0, -3), which is the y-intercept, and has a slope of -1/2. This information allows us to directly apply the slope-intercept form. Substituting the given values, we have m = -1/2 and b = -3. Therefore, the equation of the line in slope-intercept form is y = (-1/2)x - 3. This equation provides a clear and concise representation of the line, making it easy to visualize and analyze its behavior. The negative slope indicates that the line slopes downward from left to right, and the y-intercept at -3 tells us where the line crosses the vertical axis. Further manipulation of this equation can lead to other forms, such as the point-slope form or the general form, depending on the specific application or desired representation.

Leveraging the Point-Slope Form

Another powerful tool for finding the equation of a line is the point-slope form. This form is particularly useful when you know a point on the line and its slope, but not necessarily the y-intercept. The point-slope form is expressed as y - y1 = m(x - x1), where m is the slope and (x1, y1) is a known point on the line. This form is derived from the definition of slope, which is the change in y divided by the change in x. By rearranging this definition, we arrive at the point-slope form. The advantage of the point-slope form is that it directly incorporates a specific point on the line, making it easy to construct the equation. In our problem, we are given the point (0, -3) and the slope -1/2. We can directly substitute these values into the point-slope form. Let (x1, y1) = (0, -3) and m = -1/2. Plugging these values into the formula, we get y - (-3) = (-1/2)(x - 0). Simplifying this equation, we have y + 3 = (-1/2)x. This equation is a valid representation of the line, and we can further manipulate it to obtain the slope-intercept form or the general form if desired. The point-slope form highlights the relationship between the slope, a specific point, and the variables x and y, providing a flexible approach to finding the equation of a line. This method is especially beneficial when dealing with problems where the y-intercept is not immediately apparent, but a different point on the line is known.

Converting to Slope-Intercept Form

Having utilized the point-slope form, the next logical step is to convert the equation into the familiar slope-intercept form, y = mx + b. This form is highly advantageous because it explicitly reveals the slope (m) and the y-intercept (b) of the line, making it easier to visualize and analyze its properties. Our equation from the point-slope form is y + 3 = (-1/2)x. To convert this to slope-intercept form, we need to isolate y on one side of the equation. This is achieved by subtracting 3 from both sides of the equation. Performing this operation, we get y = (-1/2)x - 3. This equation is now in the slope-intercept form. We can readily identify the slope as m = -1/2 and the y-intercept as b = -3. This confirms our initial understanding of the line's characteristics. The negative slope signifies that the line slopes downward from left to right, and the y-intercept at -3 indicates the point where the line intersects the y-axis. The slope-intercept form provides a clear and concise representation of the linear relationship, facilitating further analysis and application. This conversion process demonstrates the flexibility of linear equation forms and how they can be manipulated to reveal specific information about the line.

Expressing the Equation in General Form

While the slope-intercept form is widely used and easily interpretable, another important form for linear equations is the general form. The general form is expressed as Ax + By + C = 0, where A, B, and C are constants, and A is typically a non-negative integer. The general form is particularly useful in situations where you need to represent linear equations in a standardized format, such as in systems of equations or linear programming problems. To convert our equation from slope-intercept form, y = (-1/2)x - 3, to general form, we need to rearrange the terms so that all variables and constants are on one side of the equation, set equal to zero. First, let's eliminate the fraction by multiplying the entire equation by 2, which gives us 2y = -x - 6. Next, we add x to both sides to get x + 2y = -6. Finally, we add 6 to both sides to obtain x + 2y + 6 = 0. This equation is now in the general form, where A = 1, B = 2, and C = 6. The general form is a compact and versatile representation of the linear equation. While it doesn't directly reveal the slope and y-intercept as clearly as the slope-intercept form, it provides a standardized format that is useful in various mathematical contexts. Understanding how to convert between different forms of linear equations is essential for solving a wide range of problems.

Conclusion

In summary, we have successfully found the equation of the line that passes through the point (0, -3) and has a slope of -1/2. We explored three different forms of linear equations: the slope-intercept form, the point-slope form, and the general form. We started by recognizing that the given point (0, -3) is the y-intercept and directly applied the slope-intercept form to obtain y = (-1/2)x - 3. We then demonstrated how to use the point-slope form, y - y1 = m(x - x1), and converted the resulting equation to slope-intercept form. Finally, we converted the slope-intercept form to the general form, Ax + By + C = 0, which resulted in the equation x + 2y + 6 = 0. Understanding these different forms and how to convert between them is a crucial skill in algebra and beyond. Linear equations are fundamental tools in mathematics and have wide-ranging applications in various fields, including physics, engineering, economics, and computer science. By mastering the concepts and techniques discussed in this article, you will be well-equipped to tackle a variety of problems involving linear relationships.