Slope & Y-intercept: Graphing 4x + 3y = -3

Understanding how to graph linear equations is a fundamental skill in algebra. One common form of a linear equation is 4x + 3y = -3. To effectively graph this equation, we need to identify its slope and y-intercept. These two key features provide us with enough information to plot the line on a coordinate plane accurately. Let's break down the process step-by-step. First, we'll focus on rearranging the equation into slope-intercept form. This form, y = mx + b, is particularly useful because 'm' directly represents the slope of the line and 'b' represents the y-coordinate of the y-intercept. By converting our given equation into this form, we can easily extract the values we need for graphing. Next, we'll pinpoint the y-intercept. The y-intercept is the point where the line crosses the y-axis. Once we have both the slope and y-intercept, we can use them to plot points on the graph and draw the line. The slope tells us how steep the line is and in what direction it's oriented (increasing or decreasing). The y-intercept gives us a starting point on the y-axis from which to apply the slope. In the following sections, we'll go through each step in detail, providing clear explanations and examples to ensure a solid understanding of the concepts involved. Understanding the relationship between a linear equation, its slope, and its y-intercept is crucial not only for graphing but also for solving various algebraic problems and modeling real-world situations. This skill provides a foundation for more advanced mathematical concepts and applications.

Step 1: Convert to Slope-Intercept Form (y = mx + b)

To find the slope and y-intercept of the line represented by the equation 4x + 3y = -3, we first need to convert it into slope-intercept form, which is y = mx + b. This form explicitly reveals the slope (m) and the y-intercept (b) of the line. Let's walk through the algebraic steps to achieve this transformation. We start with the original equation: 4x + 3y = -3. Our goal is to isolate y on one side of the equation. To do this, we first subtract 4x from both sides of the equation. This gives us: 3y = -4x - 3. Now, to completely isolate y, we need to divide both sides of the equation by 3. This results in: y = (-4/3)x - 1. Now the equation is in slope-intercept form. By comparing this to y = mx + b, we can directly identify the slope and y-intercept. The coefficient of x, which is -4/3, is the slope (m). The constant term, which is -1, is the y-intercept (b). Therefore, we have successfully transformed the original equation into slope-intercept form, making it easy to identify the key parameters needed for graphing the line. Understanding how to manipulate equations into different forms is a crucial skill in algebra. It allows us to extract specific information and solve problems more efficiently. In this case, converting to slope-intercept form allowed us to easily determine the slope and y-intercept, which are essential for graphing the line. The ability to rearrange equations and isolate variables is a fundamental building block for more advanced mathematical concepts.

Step 2: Identify the Slope and Y-intercept

Now that our equation is in slope-intercept form (y = (-4/3)x - 1), identifying the slope and y-intercept is straightforward. As we discussed earlier, the slope (m) is the coefficient of the x term, and the y-intercept (b) is the constant term. In this case, the slope, m, is -4/3. This means that for every 3 units we move to the right on the graph, the line goes down 4 units. A negative slope indicates that the line is decreasing, or sloping downwards, from left to right. The y-intercept, b, is -1. This tells us that the line crosses the y-axis at the point (0, -1). The y-intercept is the point where x = 0, and it's a crucial reference point for graphing the line. Understanding the meaning of the slope and y-intercept is essential for interpreting linear equations. The slope describes the rate of change of the line, while the y-intercept provides a starting point on the y-axis. Together, they give us a complete picture of the line's position and orientation on the coordinate plane. This information is not only useful for graphing but also for analyzing and modeling real-world relationships. For example, in a linear cost function, the slope might represent the cost per unit, and the y-intercept might represent the fixed costs. By understanding the slope and y-intercept, we can gain valuable insights into the relationship between the variables being modeled.

Step 3: Graph the Line

With the slope (m = -4/3) and y-intercept (b = -1) in hand, we can now graph the line 4x + 3y = -3. Here’s how to do it: First, plot the y-intercept. Since the y-intercept is -1, we place a point at (0, -1) on the coordinate plane. This is our starting point. Next, use the slope to find another point on the line. Remember that the slope is rise over run. In this case, the slope is -4/3, which means for every 3 units we move to the right (run), we move 4 units down (rise). Starting from the y-intercept (0, -1), move 3 units to the right to (3, -1). Then, move 4 units down from (3, -1) to (3, -5). This gives us a second point on the line: (3, -5). Now that we have two points, (0, -1) and (3, -5), we can draw a straight line through them. This line represents the equation 4x + 3y = -3. Extend the line beyond the two points to cover the entire graph. It's always a good idea to check your graph by plugging in a few additional x-values into the equation and verifying that the corresponding y-values fall on the line you've drawn. This helps ensure that your graph is accurate. Graphing linear equations is a fundamental skill in algebra and has numerous applications in various fields. By understanding the relationship between the equation, slope, and y-intercept, you can accurately represent linear relationships visually and use them to solve problems. Remember that the slope determines the steepness and direction of the line, while the y-intercept provides a starting point on the y-axis. With practice, you'll become proficient at graphing linear equations and using them to model real-world scenarios.

In conclusion, by converting the equation 4x + 3y = -3 to slope-intercept form, we successfully identified the slope as -4/3 and the y-intercept as -1. Using these values, we were able to accurately graph the line. Remember to start by plotting the y-intercept and then use the slope to find additional points on the line. Mastering this technique is essential for understanding linear equations and their graphical representations. For further learning, visit Khan Academy's Linear Equations and Graphs section to deepen your understanding: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:linear-equations-graphs