Equation Of A Line With Y-intercept -4 And Slope -1/4

Jul 14, 2025 by ADMIN 54 views

In the realm of mathematics, specifically linear equations, determining the equation of a line is a fundamental concept. This article delves into the process of finding the equation of a line given its $y$ -intercept and slope. We will focus on a specific example where the $y$ -intercept is -4 and the slope is -1/4. Understanding how to derive this equation is crucial for various applications, from graphing linear functions to solving real-world problems involving rates of change. This article will provide a detailed, step-by-step explanation, ensuring that even those new to the concept can grasp the underlying principles. Let's embark on this mathematical journey and unravel the equation that defines this unique line.

Understanding the Slope-Intercept Form

The slope-intercept form is a cornerstone in the study of linear equations. It provides a clear and concise way to represent the relationship between the variables x and y in a linear function. The slope-intercept form is expressed as:

y = mx + b

Where:

y represents the dependent variable (typically plotted on the vertical axis).
x represents the independent variable (typically plotted on the horizontal axis).
m represents the slope of the line, indicating its steepness and direction. It quantifies the change in y for every unit change in x.
b represents the $y$ -intercept, which is the point where the line crosses the $y$ -axis. It's the value of y when x is equal to 0.

The beauty of the slope-intercept form lies in its simplicity and the immediate information it provides about the line. By simply looking at the equation, we can identify the slope and the $y$ -intercept, which are essential for graphing the line and understanding its behavior. The slope, m, is often referred to as "rise over run," representing the vertical change (rise) divided by the horizontal change (run) between any two points on the line. A positive slope indicates an increasing line (from left to right), while a negative slope indicates a decreasing line. The y-intercept, b, provides a fixed point (0, b) through which the line passes. This form is not just a mathematical construct; it's a powerful tool for visualizing and analyzing linear relationships in various fields, from physics and engineering to economics and data analysis.

Identifying the Given Information

Before we can construct the equation of the line, we must first clearly identify the given information. In this problem, we are provided with two crucial pieces of information:

The $y$ -intercept: The line intersects the $y$ -axis at the point where $y = -4$ . This means the value of b in our slope-intercept form is -4.
The slope: The line has a slope of $- rac{1}{4}$ . This means for every 4 units we move to the right along the x-axis, the line descends 1 unit along the y-axis. Therefore, the value of m in our slope-intercept form is $- rac{1}{4}$ .

Having these two pieces of information is sufficient to define the line uniquely. The $y$ -intercept anchors the line to a specific point on the $y$ -axis, while the slope dictates its orientation and steepness. The slope of $- rac{1}{4}$ indicates that the line slopes downwards from left to right, a gentle decline compared to a steeper negative slope. The $y$ -intercept of -4 tells us that the line passes through the point (0, -4). These two values are the key ingredients we need to formulate the equation of the line. Recognizing and extracting these values from the problem statement is the first essential step in solving this type of problem. This careful identification ensures that we are working with the correct parameters and sets the stage for a successful application of the slope-intercept form.

Substituting Values into the Slope-Intercept Form

Now that we have identified the slope (m) and the $y$ -intercept (b), the next step is to substitute these values into the slope-intercept form equation: $y = mx + b$ . This substitution is a direct application of the formula, replacing the generic symbols m and b with the specific values we have for our line.

We know that:

$m = - rac{1}{4}$ (the slope)
$b = -4$ (the $y$ -intercept)

Substituting these values into the equation, we get:

y = ext{-} rac{1}{4}x + (-4)

This equation now represents the specific line we are interested in. It describes the relationship between x and y for all points that lie on this line. The substitution process is a crucial step in bridging the gap between the general form of a linear equation and the specific equation that defines our line. It's a simple yet powerful technique that allows us to tailor the general equation to fit the unique characteristics of our line, namely its slope and $y$ -intercept. This substituted equation is the almost final form, needing only a slight simplification to reach its most elegant and readily usable state. The act of substitution transforms the abstract into the concrete, giving us a tangible representation of the line we are trying to define.

Simplifying the Equation

After substituting the values of the slope and $y$ -intercept into the slope-intercept form, we arrive at the equation: $y = - rac{1}{4}x + (-4)$ . To complete the process and present the equation in its most standard form, we need to simplify it. This simplification primarily involves dealing with the addition of a negative number.

In mathematics, adding a negative number is equivalent to subtraction. Therefore, we can rewrite the equation as:

y = - rac{1}{4}x - 4

This is the simplified equation of the line with a $y$ -intercept of -4 and a slope of $- rac{1}{4}$ . This form is not only more concise but also easier to interpret and use for various purposes, such as graphing the line or finding specific points on the line. The simplification step is important because it ensures that the equation is presented in a conventional and readily recognizable format. It removes any unnecessary clutter and makes the equation more accessible. The final equation, $y = - rac{1}{4}x - 4$ , is the culmination of our efforts, a clear and unambiguous representation of the line we set out to define. This equation encapsulates all the information we were given initially and presents it in a form that is both mathematically sound and practically useful.

Final Equation and its Significance

The final equation of the line with a $y$ -intercept of -4 and a slope of $- rac{1}{4}$ is:

y = - rac{1}{4}x - 4

This equation holds significant information about the line it represents. It tells us that for every increase of 4 units in the x-direction, the y-value decreases by 1 unit. The line intersects the y-axis at the point (0, -4). This equation is not just a string of symbols; it's a powerful statement about the relationship between x and y for all points on this line.

The significance of this equation extends beyond the realm of pure mathematics. Linear equations are used extensively in various fields, including physics, engineering, economics, and computer science. They can model a wide range of phenomena, from the motion of objects to the growth of populations to the relationship between supply and demand. Understanding how to derive and interpret linear equations is a valuable skill that can be applied in many different contexts. The ability to find the equation of a line given its slope and $y$ -intercept is a fundamental building block for more advanced mathematical concepts and real-world applications. This final equation, therefore, is not just the answer to a specific problem; it's a gateway to a deeper understanding of the world around us. It's a testament to the power of mathematics to describe and predict patterns and relationships in a precise and elegant way.

In conclusion, we have successfully found the equation of the line with a $y$ -intercept of -4 and a slope of $- rac{1}{4}$ by utilizing the slope-intercept form and performing the necessary substitutions and simplifications. This process highlights the fundamental principles of linear equations and their applications in various fields.