Equation Of A Line Slope -4 Passing Through Point (-3, -8)

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In the realm of mathematics, linear equations form the bedrock of numerous concepts and applications. Understanding how to derive the equation of a line given specific parameters is a fundamental skill. This article delves into a step-by-step approach to finding the equation of a line when the slope and a point on the line are known. We will specifically tackle the problem of finding the equation of a line with a slope of -4 that passes through the point (-3, -8). By mastering this process, you'll gain a valuable tool for tackling more complex mathematical challenges.

The problem we aim to solve is this: given a slope m = -4 and a point (-3, -8), determine the equation of the line. This seemingly simple task unveils a powerful method applicable in various mathematical contexts. Let's embark on this journey by first exploring the fundamental concepts of linear equations and their representations.

At its core, a linear equation represents a straight line on a coordinate plane. There are several forms in which a linear equation can be expressed, but one of the most commonly used and intuitive forms is the slope-intercept form. This form provides a direct way to visualize the line's characteristics, namely its slope and y-intercept. 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
  • b represents the y-intercept (the point where the line crosses the y-axis)

The slope, denoted by m, quantifies the steepness and direction of the line. It signifies the rate of change in y for a unit change in x. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. The magnitude of the slope reflects the steepness; a larger absolute value signifies a steeper line. In our problem, we are given a slope of -4, indicating a line that slopes downwards from left to right.

The y-intercept, denoted by b, is the point where the line intersects the y-axis. It is the value of y when x is equal to 0. The y-intercept provides a fixed point of reference for the line's position on the coordinate plane. To determine the equation of a line, we need to find both the slope (m) and the y-intercept (b). We are already given the slope in our problem, so the next step is to find the y-intercept using the given point.

While the slope-intercept form is useful for visualizing the line, the point-slope form is a more direct tool for finding the equation of a line when given a point and the slope. The point-slope form is expressed as:

y - y1 = m(x - x1)

where:

  • m represents the slope of the line
  • (x1, y1) represents a known point on the line

This form directly incorporates the given information – the slope and a point – into the equation. It states that the difference in the y-coordinates between any point on the line (y) and the given point (y1) is equal to the slope (m) times the difference in the x-coordinates between the same point (x) and the given point (x1). By substituting the given values into the point-slope form, we can derive the equation of the line.

In our problem, we have m = -4 and the point (-3, -8). We can directly substitute these values into the point-slope form:

y - (-8) = -4(x - (-3))

This equation represents the line with the given slope and passing through the given point. However, to express the equation in a more standard form, we need to simplify it further. The next step involves algebraic manipulation to convert the equation into slope-intercept form or standard form.

Now, let's apply the point-slope form to our specific problem. We are given a slope m = -4 and a point (-3, -8). Substituting these values into the point-slope form, we get:

y - (-8) = -4(x - (-3))

Simplifying the equation, we have:

y + 8 = -4(x + 3)

To further simplify, we distribute the -4 on the right side:

y + 8 = -4x - 12

Our goal is to express the equation in slope-intercept form (y = mx + b), so we need to isolate y. To do this, we subtract 8 from both sides of the equation:

y = -4x - 12 - 8

Combining the constants, we get:

y = -4x - 20

This is the equation of the line in slope-intercept form. We have successfully found the equation of the line with a slope of -4 that passes through the point (-3, -8). The y-intercept of this line is -20, which means the line crosses the y-axis at the point (0, -20). Let's solidify our understanding by examining the implications of this equation and comparing it to the given options.

To ensure the accuracy of our solution, we need to verify that the derived equation, y = -4x - 20, indeed represents a line with a slope of -4 and passes through the point (-3, -8). The slope is readily apparent from the equation itself; the coefficient of x is -4, which confirms the slope. To verify that the line passes through the point (-3, -8), we substitute the x and y coordinates of the point into the equation:

-8 = -4(-3) - 20

Simplifying the right side:

-8 = 12 - 20

-8 = -8

The equation holds true, confirming that the point (-3, -8) lies on the line represented by y = -4x - 20. This verification step is crucial to ensure that our solution is correct. Now, let's compare our solution to the given options:

A. y = -4x + 20 B. y = -4x - 20 C. x = -4y + 20 D. x = -4y - 20

Our derived equation, y = -4x - 20, matches option B. The other options either have an incorrect y-intercept (option A) or are not in the standard form of a linear equation (options C and D). Options C and D are expressed with x as a function of y, which represents a horizontal line if the coefficient of y were zero, or an inverse relationship if not. Therefore, option B is the correct answer.

In this article, we have successfully found the equation of a line with a slope of -4 that passes through the point (-3, -8). We began by understanding the fundamental concepts of linear equations, specifically the slope-intercept form (y = mx + b) and the point-slope form (y - y1 = m(x - x1)). We then applied the point-slope form to derive the equation, simplified it into slope-intercept form, and verified our solution by substituting the given point into the equation.

The equation y = -4x - 20 represents the line that satisfies the given conditions. This exercise demonstrates the power of the point-slope form in determining the equation of a line when a point and the slope are known. Mastering this technique is essential for tackling more advanced mathematical problems involving linear equations and their applications. By understanding the relationship between the slope, y-intercept, and points on a line, you can confidently solve a wide range of problems in algebra, geometry, and beyond. The ability to manipulate and interpret linear equations is a cornerstone of mathematical proficiency.