Equation Of A Line Slope Intercept Form Through (1 -1) Slope -5

In the realm of mathematics, the concept of a line stands as a fundamental building block. Lines, with their inherent simplicity and predictability, form the basis for numerous mathematical models and real-world applications. Understanding the properties of lines, such as their slope and intercepts, is crucial for solving a wide range of problems, from determining the trajectory of a projectile to modeling the growth of a population. In this comprehensive exploration, we embark on a journey to unravel the equation of a specific line, one that gracefully traverses the point (1, -1) and possesses a slope of -5. Our ultimate goal is to express this equation in the elegant slope-intercept form, a form that provides immediate insights into the line's characteristics.

Understanding the Slope-Intercept Form

The slope-intercept form, a cornerstone in the study of linear equations, offers a clear and concise way to represent the equation of a line. This form, expressed as y = mx + b, elegantly encapsulates the two essential properties of a line: its slope (m) and its y-intercept (b). The slope, denoted by 'm', quantifies the steepness of the line, indicating the rate at which the line rises or falls as we move along the x-axis. A positive slope signifies an upward slant, while a negative slope indicates a downward trend. The y-intercept, represented by 'b', marks the point where the line intersects the y-axis, providing a fixed reference point for the line's position in the coordinate plane. The slope-intercept form's simplicity and interpretability make it an invaluable tool for analyzing and manipulating linear equations.

Decoding the Slope: A Measure of Steepness

The slope, a fundamental concept in understanding lines, serves as a measure of the line's steepness and direction. Mathematically, the slope is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two distinct points on the line. This ratio, often denoted by the letter 'm', provides a numerical representation of how much the line rises or falls for every unit of horizontal movement. A positive slope indicates an upward slant, meaning that the line rises as we move from left to right along the x-axis. Conversely, a negative slope signifies a downward slant, with the line falling as we progress horizontally. A slope of zero corresponds to a horizontal line, while an undefined slope represents a vertical line. The magnitude of the slope reflects the steepness of the line, with larger absolute values indicating steeper inclines or declines.

The Y-Intercept: A Line's Anchor Point

The y-intercept, a crucial element in defining a line, marks the point where the line intersects the y-axis. This point, represented by the coordinates (0, b), provides a fixed reference for the line's position in the coordinate plane. The y-intercept is particularly significant in the slope-intercept form of a linear equation (y = mx + b), where 'b' directly corresponds to the y-coordinate of this intersection point. The y-intercept serves as a starting point for graphing the line and provides valuable information about the line's behavior. For instance, if the y-intercept is positive, the line crosses the y-axis above the origin, while a negative y-intercept indicates an intersection below the origin. Understanding the y-intercept is essential for accurately interpreting and manipulating linear equations.

The Point-Slope Form: A Stepping Stone

Before we can express the equation of our line in slope-intercept form, we must first utilize the point-slope form, another valuable representation of linear equations. The point-slope form, given by y - y1 = m(x - x1), elegantly incorporates the slope (m) and a specific point (x1, y1) that the line passes through. This form is particularly useful when we know the slope of a line and a single point on it, as is the case in our problem. By substituting the given slope and point into the point-slope form, we can obtain an equation that accurately represents the line. This equation, while not yet in slope-intercept form, serves as a crucial stepping stone in our journey to unveil the line's ultimate equation.

Deriving the Point-Slope Form

The point-slope form, a powerful tool in linear algebra, provides a direct way to express the equation of a line when we know its slope and a single point it passes through. The derivation of this form stems from the fundamental definition of slope. Recall that the slope (m) of a line is defined as the ratio of the vertical change (Δy) to the horizontal change (Δx) between any two points on the line. Let (x1, y1) be the known point on the line, and let (x, y) represent any other arbitrary point on the same line. Then, the slope can be expressed as m = (y - y1) / (x - x1). To obtain the point-slope form, we simply multiply both sides of this equation by (x - x1), resulting in the equation y - y1 = m(x - x1). This equation, known as the point-slope form, elegantly captures the relationship between the slope, a known point, and any other point on the line.

Applying the Point-Slope Form to Our Problem

In our specific problem, we are given that the line passes through the point (1, -1) and has a slope of -5. We can readily apply the point-slope form by substituting these values into the equation y - y1 = m(x - x1). Let (x1, y1) be (1, -1) and m be -5. Substituting these values, we get: y - (-1) = -5(x - 1). This equation, while technically correct, is not yet in the desired slope-intercept form. Our next step is to manipulate this equation algebraically to transform it into the familiar y = mx + b format.

Substituting the Given Values

To substitute the given values into the point-slope form, we replace the variables y1, x1, and m with their respective values. In our case, we have the point (1, -1), which means x1 = 1 and y1 = -1. We also know that the slope, m, is -5. Substituting these values into the point-slope form equation, y - y1 = m(x - x1), we get: y - (-1) = -5(x - 1). This substitution is a crucial step in converting the given information into a usable equation that represents the line. The equation now encapsulates the line's slope and its passage through the specified point. However, to fully understand the line's behavior and easily graph it, we need to transform this equation into the slope-intercept form.

Transforming to Slope-Intercept Form

To transform the equation from point-slope form to slope-intercept form, we must perform a series of algebraic manipulations. Our goal is to isolate 'y' on one side of the equation, expressing it in terms of 'x' and a constant. Starting with the equation y - (-1) = -5(x - 1), we first simplify the left side by removing the double negative, resulting in y + 1 = -5(x - 1). Next, we distribute the -5 on the right side, giving us y + 1 = -5x + 5. Finally, we subtract 1 from both sides to isolate 'y', yielding the equation y = -5x + 4. This equation is now in slope-intercept form, revealing the slope (m = -5) and the y-intercept (b = 4) of the line.

Isolating 'Y': The Key to Slope-Intercept Form

Isolating 'y' is the central step in converting an equation to slope-intercept form. This process involves performing algebraic operations on both sides of the equation to get 'y' by itself on one side. In our case, we started with the equation y + 1 = -5x + 5. To isolate 'y', we need to eliminate the '+ 1' on the left side. This is achieved by subtracting 1 from both sides of the equation. Subtracting 1 from both sides maintains the equality and allows us to simplify the equation. The result of this subtraction is y = -5x + 4, where 'y' is now isolated. This final equation is in slope-intercept form, allowing us to easily identify the slope and y-intercept of the line.

The Final Equation: Y = -5x + 4

After our algebraic journey, we have arrived at the equation in slope-intercept form for the line passing through (1, -1) with a slope of -5: y = -5x + 4. This equation elegantly encapsulates the line's characteristics. The slope, -5, indicates that the line descends steeply as we move from left to right, dropping 5 units vertically for every 1 unit of horizontal movement. The y-intercept, 4, reveals that the line intersects the y-axis at the point (0, 4). This equation not only provides a concise representation of the line but also allows us to readily graph it and make predictions about its behavior.

Interpreting the Equation: Slope and Y-Intercept Revealed

The equation y = -5x + 4 provides a wealth of information about the line it represents. The coefficient of 'x', which is -5, directly corresponds to the slope of the line. This negative slope tells us that the line is decreasing, meaning it slopes downwards as we move from left to right. The magnitude of the slope, 5, indicates the steepness of the line; for every 1 unit we move horizontally, the line drops 5 units vertically. The constant term, 4, represents the y-intercept, the point where the line crosses the y-axis. This means the line intersects the y-axis at the point (0, 4). By simply examining the equation, we can immediately visualize the line's direction, steepness, and position in the coordinate plane. This interpretability is one of the key advantages of the slope-intercept form.

In conclusion, by meticulously applying the point-slope form and skillfully transforming it into the slope-intercept form, we have successfully unveiled the equation of the line passing through the point (1, -1) with a slope of -5. The equation, y = -5x + 4, stands as a testament to the power of algebraic manipulation and the elegance of linear equations.