Graphing A Line With Slope -2/3 Through Point (3, 5) A Step-by-Step Guide

In the realm of mathematics, particularly in coordinate geometry, understanding how to graph a line given its slope and a point it passes through is a fundamental skill. This article will delve into the process of graphing a line with a slope of $-\frac{2}{3}$ that passes through the point $(3, 5)$. We will explore the concepts of slope, point-slope form, and the step-by-step methodology to accurately plot this line on a coordinate plane. Grasping these concepts is crucial for various applications in mathematics, physics, engineering, and other fields where linear relationships are prevalent. Before diving into the specifics of our example, let’s first establish a solid understanding of the core principles involved. We will discuss the significance of slope as a measure of a line's steepness and direction, the point-slope form as a powerful tool for defining a line, and the practical steps for translating this information into a visual representation on a graph. Our aim is to provide a comprehensive guide that not only demonstrates the solution but also illuminates the underlying mathematical concepts, ensuring a deeper and more lasting understanding. The ability to graph lines accurately is a cornerstone of mathematical literacy, enabling us to visualize and analyze linear relationships in a clear and intuitive manner. This skill is not only essential for academic pursuits but also for real-world applications where linear models are used to represent various phenomena. So, let's embark on this journey of graphical exploration and unlock the power of visualizing linear equations.

Before we graph the line, let's understand the slope and point-slope form. The slope of a line, often denoted by m, quantifies its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. A positive slope indicates that the line rises as we move from left to right, while a negative slope indicates that the line falls. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line. In our case, the slope is given as $-\frac{2}{3}$, which means for every 3 units we move horizontally, the line falls 2 units vertically. This negative slope tells us that the line will be decreasing as we move from left to right on the graph. The point-slope form is a powerful tool for defining a line when we know a point it passes through and its slope. The point-slope form equation is expressed as: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line, and m is the slope. This form is particularly useful because it directly incorporates the given information (a point and a slope) into the equation of the line. To use the point-slope form effectively, we simply substitute the coordinates of the given point and the value of the slope into the equation. Once we have the equation in point-slope form, we can manipulate it further to obtain the slope-intercept form ($y = mx + b$), which is often more convenient for graphing. Understanding the relationship between the point-slope form and the slope-intercept form is crucial for a comprehensive grasp of linear equations. The point-slope form allows us to construct the equation directly from the given information, while the slope-intercept form provides a clear visual representation of the line's slope and y-intercept. By mastering these concepts, we can confidently tackle a wide range of graphing problems and apply them to real-world scenarios where linear relationships are present.

In our specific problem, we are given a slope of $-\frac{2}{3}$ and a point $(3, 5)$ through which the line passes. To apply the point-slope form, we substitute these values into the equation $y - y_1 = m(x - x_1)$. Here, $m = -\frac{2}{3}$, $x_1 = 3$, and $y_1 = 5$. Substituting these values, we get: $y - 5 = -\frac{2}{3}(x - 3)$. This equation represents the line we want to graph in point-slope form. While this form is perfectly valid, it is often helpful to convert it to slope-intercept form ($y = mx + b$) for easier graphing. To do this, we distribute the $-\frac{2}{3}$ on the right side of the equation and then isolate y. First, distribute the slope: $y - 5 = -\frac{2}{3}x + 2$. Next, add 5 to both sides to isolate y: $y = -\frac{2}{3}x + 2 + 5$, which simplifies to $y = -\frac{2}{3}x + 7$. Now we have the equation in slope-intercept form, where the slope is $-\frac{2}{3}$ and the y-intercept is 7. This form makes it very clear where the line intersects the y-axis and how steep it is. The ability to convert between point-slope form and slope-intercept form is a valuable skill in linear algebra. It allows us to easily move between different representations of the same line, choosing the form that is most convenient for the task at hand. In this case, the slope-intercept form is particularly useful for graphing because it directly tells us the y-intercept, which is one point on the line, and the slope, which we can use to find other points. By mastering this process, we can confidently handle a wide variety of linear equation problems.

Now that we have the equation of the line in slope-intercept form ($y = -\frac{2}{3}x + 7$), we can proceed with the graphing process. The first step is to plot the y-intercept. The y-intercept is the point where the line crosses the y-axis, and in our equation, it is 7. This means the line passes through the point $(0, 7)$. Locate this point on the coordinate plane and mark it clearly. Next, we use the slope to find another point on the line. The slope is $-\frac{2}{3}$, which means for every 3 units we move to the right (run), we move 2 units down (rise, but in this case, it's a fall since the slope is negative). Starting from the y-intercept $(0, 7)$, move 3 units to the right along the x-axis, which brings us to $x = 3$. Then, move 2 units down along the y-axis, which brings us to $y = 5$. This gives us a second point on the line: $(3, 5)$. This point was actually given to us in the original problem, which confirms that our calculations are correct. Now that we have two points on the line, $(0, 7)$ and $(3, 5)$, we can draw a straight line through them. Use a ruler or straightedge to ensure the line is accurate and extends beyond the two points. This line represents the graphical representation of the equation $y = -\frac{2}{3}x + 7$, or equivalently, the line with a slope of $-\frac{2}{3}$ passing through the point $(3, 5)$. Remember to label the line with its equation to clearly identify it on the graph. Accurate graphing requires careful attention to detail. Make sure to use a consistent scale on both axes and plot the points precisely. The ability to graph lines accurately is a fundamental skill in mathematics, and with practice, it becomes a straightforward and intuitive process.

After plotting the two points and drawing the line, it is crucial to visually inspect the graph to ensure it aligns with our expectations. Our slope is $-\frac{2}{3}$ which is negative, so the line should be decreasing as we move from left to right. If the line appears to be increasing, it indicates a potential error in our calculations or graphing process. Visually, the line should pass through the point $(3, 5)$, which was given in the problem statement. If the drawn line does not pass through this point, it suggests an inaccuracy in the plotting or equation derivation. Furthermore, we can choose another point on the line and substitute its coordinates into the equation $y = -\frac{2}{3}x + 7$ to verify that it satisfies the equation. For instance, let's consider the point $(6, 3)$, which appears to lie on the line. Substituting $x = 6$ into the equation, we get: $y = -\frac{2}{3}(6) + 7 = -4 + 7 = 3$. Since the calculated y-value matches the y-coordinate of the point, this confirms that the point lies on the line and our graph is likely accurate. The visual representation of a line provides a powerful way to understand and analyze linear relationships. By observing the steepness and direction of the line, we can gain insights into the relationship between the variables it represents. In this case, the line with a slope of $-\frac{2}{3}$ represents a linear relationship where for every increase of 3 units in x, y decreases by 2 units. This visual understanding is invaluable in various applications, such as modeling real-world scenarios, interpreting data, and solving problems involving linear equations. The process of graphing and verifying the line reinforces our understanding of the relationship between algebraic equations and their geometric representations.

While we primarily used the point-slope and slope-intercept forms to graph the line, there are alternative methods we could have employed. One such method is to find two points directly by choosing two different x-values, substituting them into the equation $y = -\frac{2}{3}x + 7$, and calculating the corresponding y-values. For example, we could have chosen $x = 0$ and $x = 3$, which would have given us the points $(0, 7)$ and $(3, 5)$, as we already found. Another approach is to use the slope and a single point to find other points iteratively. Starting from $(3, 5)$, we could move 3 units to the right and 2 units down (following the slope of $-\frac{2}{3}$) to find another point, and so on. These alternative methods provide flexibility in graphing lines and can be particularly useful in different situations. In conclusion, we have successfully graphed the line with a slope of $-\frac{2}{3}$ passing through the point $(3, 5)$. We accomplished this by understanding the concepts of slope and point-slope form, converting the equation to slope-intercept form, plotting the y-intercept, using the slope to find another point, and drawing a line through these points. We also emphasized the importance of visual verification and discussed alternative graphing methods. This exercise demonstrates the fundamental principles of graphing linear equations, which are essential for various mathematical and real-world applications. By mastering these skills, we can confidently visualize and analyze linear relationships, making informed decisions and solving problems effectively. Graphing lines is not just a mathematical exercise; it is a powerful tool for understanding and representing the world around us.