Graphing The Line Through (7, -4) And (-6, -4) And Determining Its Equation
In mathematics, understanding how to graph lines and determine their equations is a fundamental skill. This article will guide you through the process of graphing a line that passes through the points (7, -4) and (-6, -4) and then determining the equation of that line. This exercise will reinforce your understanding of coordinate geometry, slope, and linear equations.
Understanding Coordinate Geometry
Before diving into the specifics, it's crucial to understand the basics of coordinate geometry. The Cartesian plane, also known as the coordinate plane, is a two-dimensional plane formed by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Points on this plane are represented by ordered pairs (x, y), where x is the horizontal distance from the origin (0, 0), and y is the vertical distance from the origin. Understanding how to plot points on this plane is the first step in graphing lines and other functions.
When we talk about graphing a line, we are essentially visualizing the set of all points that satisfy a particular linear equation. Linear equations are equations that, when graphed, form a straight line. The most common form of a linear equation is the slope-intercept form, which is written 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). Another important concept is the slope of a line, which measures the steepness and direction of the line. The slope is often described as "rise over run," which is the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. This value is crucial for determining the equation of the line once we have graphed it or identified two points on it.
Step 1: Plotting the Points (7, -4) and (-6, -4)
The first step in graphing the line is to plot the given points (7, -4) and (-6, -4) on the coordinate plane. To plot the point (7, -4), start at the origin (0, 0), move 7 units to the right along the x-axis, and then move 4 units down along the y-axis. Mark this point clearly on your graph. Similarly, to plot the point (-6, -4), start at the origin, move 6 units to the left along the x-axis, and then move 4 units down along the y-axis. Mark this point as well. Ensure that your axes are properly labeled, and each point is accurately placed. This precise plotting is essential for accurately graphing the line and determining its equation. Misplacing the points can lead to an incorrect graph and a wrong equation.
Step 2: Drawing the Line
Once you have plotted the points (7, -4) and (-6, -4), the next step is to draw a straight line that passes through both points. Use a ruler or straightedge to ensure the line is straight and accurate. Extend the line beyond the two points to get a clear representation of the line’s trajectory. Accuracy is key here; a slightly skewed line can lead to errors in determining the equation. The line you draw visually represents all the points that satisfy the equation of the line. This visual representation is incredibly helpful in understanding the behavior of the equation and the relationship between x and y values along the line. The line should clearly pass through both plotted points, and any deviation can indicate a potential error in plotting or drawing. This graphical representation is a powerful tool for understanding linear equations.
Step 3: Determining the Slope
The slope of a line is a crucial characteristic that tells us how steep the line is and in which direction it runs. The slope (m) is defined as the change in the y-coordinate divided by the change in the x-coordinate (rise over run) between any two points on the line. Mathematically, the slope can be calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are two points on the line. In this case, our points are (7, -4) and (-6, -4). Plugging these values into the slope formula, we get: m = (-4 - (-4)) / (-6 - 7) = (0) / (-13) = 0. A slope of 0 indicates that the line is horizontal. This means that the y-value remains constant for all x-values on the line. Understanding how to calculate slope is fundamental to understanding linear equations and their graphs. The slope not only tells us the direction and steepness of the line but also plays a critical role in defining the line's equation. This calculation is a cornerstone of linear algebra and coordinate geometry.
Step 4: Finding the Equation of the Line
Now that we have the slope (m = 0), we can determine the equation of the line. There are a couple of ways to do this, but one straightforward method is to use the slope-intercept form of a linear equation, which is y = mx + b, where 'b' is the y-intercept. We already know the slope (m = 0), so our equation becomes y = 0x + b, which simplifies to y = b. To find the value of 'b', we can use either of the given points (7, -4) or (-6, -4). Let's use the point (7, -4). Substitute the y-value (-4) into the equation: -4 = b. So, the y-intercept (b) is -4. Therefore, the equation of the line is y = -4. This means that for any x-value, the y-value is always -4. Understanding the relationship between the slope, y-intercept, and the equation of a line is crucial for mastering linear equations. The equation y = -4 represents a horizontal line that intersects the y-axis at -4. This equation is a concise way to describe the line we have graphed and provides valuable information about the line's behavior and position on the coordinate plane.
Step 5: Verifying the Equation
To ensure the equation y = -4 is correct, we can verify it by substituting both points (7, -4) and (-6, -4) into the equation. For the point (7, -4), substituting y = -4 gives us -4 = -4, which is true. For the point (-6, -4), substituting y = -4 also gives us -4 = -4, which is true. Since both points satisfy the equation, we can confidently conclude that the equation of the line is indeed y = -4. This verification step is important in mathematics to ensure the accuracy of our results. Verifying the equation with the given points helps to confirm our understanding of the relationship between points, lines, and their algebraic representations. It also highlights the importance of double-checking our work to avoid errors and build confidence in our solutions.
Conclusion
In this article, we successfully graphed the line that passes through the points (7, -4) and (-6, -4) and determined its equation to be y = -4. This process involved plotting the points on the coordinate plane, drawing a line through them, calculating the slope, and using the slope and a point to find the equation of the line. By verifying the equation with both given points, we ensured the accuracy of our result. This exercise demonstrates the fundamental principles of coordinate geometry and linear equations, which are essential for further studies in mathematics. Understanding these concepts will provide a solid foundation for more advanced topics, such as systems of equations, linear transformations, and calculus. Mastering the art of graphing lines and determining their equations is a cornerstone of mathematical literacy.
