Slope-Intercept Form Equation With A Slope Of 6/7 And A Y-Intercept Of (0, -6)

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In the realm of linear equations, the slope-intercept form stands out as a fundamental and intuitive way to represent a straight line. This form, expressed as y = mx + b, provides a clear understanding of a line's characteristics, namely its slope (m) and y-intercept (b). Understanding the slope-intercept form is crucial for various mathematical applications, including graphing lines, solving systems of equations, and modeling real-world scenarios. This article delves into the intricacies of the slope-intercept form, guiding you through the process of writing equations in this form and highlighting its significance in mathematics and beyond.

Delving into Slope-Intercept Form

The slope-intercept form, y = mx + b, is a powerful tool for representing linear equations. Let's break down each component:

  • y: Represents the dependent variable, typically plotted on the vertical axis.
  • x: Represents the independent variable, typically plotted on the horizontal axis.
  • m: Denotes the slope of the line, indicating its steepness and direction. It represents the change in y for every unit change in x. A positive slope signifies an upward trend, while a negative slope indicates a downward trend. A slope of zero represents a horizontal line.
  • b: Represents the y-intercept, the point where the line intersects the y-axis. It is the value of y when x is zero.

The beauty of the slope-intercept form lies in its simplicity and interpretability. By simply looking at the equation, you can immediately identify the slope and y-intercept, which are crucial for graphing the line and understanding its behavior.

Determining the Slope

The slope, often denoted by m, quantifies the steepness and direction of a line. It is defined as the change in y divided by the change in x, often referred to as "rise over run." Mathematically, the slope can be calculated using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

where (x₁, y₁) and (x₂, y₂) are any two distinct points on the line.

A positive slope indicates that the line is increasing (going upwards) as you move from left to right, while a negative slope indicates that the line is decreasing (going downwards). A zero slope signifies a horizontal line, and an undefined slope represents a vertical line.

Understanding the concept of slope is essential for interpreting linear relationships. In real-world scenarios, the slope can represent rates of change, such as the speed of a car, the growth rate of a population, or the price change of a stock.

Identifying the y-intercept

The y-intercept, denoted by b in the slope-intercept form, is the point where the line crosses the y-axis. At this point, the x-coordinate is always zero. The y-intercept provides a starting point for graphing the line and represents the value of y when x is zero.

To find the y-intercept, you can either look at the equation in slope-intercept form (y = mx + b), where b is the y-intercept, or substitute x = 0 into the equation and solve for y. The resulting value of y is the y-intercept.

The y-intercept is a crucial parameter in many real-world applications. For example, in a linear cost function, the y-intercept might represent the fixed costs, while in a linear depreciation model, the y-intercept could represent the initial value of an asset.

Writing Equations in Slope-Intercept Form: A Step-by-Step Approach

To write the equation of a line in slope-intercept form, you need two pieces of information: the slope (m) and the y-intercept (b). Once you have these values, simply substitute them into the equation y = mx + b.

Let's consider the problem at hand: write the equation in slope-intercept form of a line with a slope of 6/7 and a y-intercept of (0, -6). We are given the slope, m = 6/7, and the y-intercept, which is the point (0, -6). This means b = -6.

Now, substitute these values into the slope-intercept form equation:

y = mx + b y = (6/7)x + (-6) y = (6/7)x - 6

Therefore, the equation of the line in slope-intercept form is y = (6/7)x - 6.

This equation represents a line that has a slope of 6/7, meaning that for every 7 units you move to the right along the x-axis, you move 6 units up along the y-axis. The line intersects the y-axis at the point (0, -6).

Additional Examples and Applications

To further solidify your understanding, let's explore some additional examples:

Example 1:

Write the equation of a line with a slope of -2 and a y-intercept of (0, 5).

Solution:

m = -2 b = 5 y = mx + b y = -2x + 5

Example 2:

Write the equation of a line that passes through the points (1, 2) and (3, 8).

Solution:

First, find the slope:

m = (8 - 2) / (3 - 1) = 6 / 2 = 3

Next, use the point-slope form of a line, y - y₁ = m(x - x₁), and substitute one of the points and the slope:

y - 2 = 3(x - 1)

Now, convert to slope-intercept form:

y - 2 = 3x - 3 y = 3x - 1

These examples demonstrate how to write equations in slope-intercept form using different given information. The slope-intercept form is widely used in various applications, including:

  • Graphing lines: The slope and y-intercept provide immediate information for plotting the line.
  • Solving systems of equations: The slope-intercept form can be used to solve systems of linear equations graphically or algebraically.
  • Modeling real-world scenarios: Linear equations in slope-intercept form can model various real-world phenomena, such as linear growth, depreciation, and cost functions.

Conclusion

The slope-intercept form, y = mx + b, is a fundamental concept in linear algebra, offering a clear and concise way to represent a straight line. By understanding the slope and y-intercept, you can readily graph lines, solve linear equations, and model real-world scenarios. This article has provided a comprehensive guide to the slope-intercept form, covering its definition, components, and applications. Mastering this form is crucial for success in mathematics and related fields.

By understanding the slope-intercept form, you gain a powerful tool for analyzing and interpreting linear relationships, which are prevalent in numerous aspects of our world. Whether you're graphing lines, solving equations, or modeling real-world phenomena, the slope-intercept form provides a valuable framework for understanding the behavior of straight lines.