Slope-Intercept Form: Rewriting 9x + 12y = -19

Let's dive into how to rewrite the equation $9x + 12y = -19$ into slope-intercept form. Guys, this is a fundamental skill in algebra, and mastering it will help you understand linear equations much better. The slope-intercept form is a way of writing linear equations that makes it super easy to identify the slope and y-intercept of the line. This form is written as $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the y-intercept. So, our goal here is to manipulate the given equation to match this format. We'll go step-by-step to make sure everyone can follow along. First, we need to isolate the $y$ term on one side of the equation. This involves moving the $x$ term to the other side. Then, we'll divide by the coefficient of $y$ to get $y$ all by itself. Once we have $y$ by itself, the equation will be in the desired $y = mx + b$ form. From there, you can easily read off the slope $m$ and the y-intercept $b$. Understanding these concepts is crucial not only for solving equations but also for graphing lines and understanding their behavior. We'll break down each step with clear explanations to make it as straightforward as possible. So, let's get started and transform $9x + 12y = -19$ into its slope-intercept form! Remember, practice makes perfect, so work through this example and try others to solidify your understanding. Slope-intercept form is your friend when it comes to linear equations!

Step-by-Step Conversion

Alright, let's get into the nitty-gritty of converting the equation $9x + 12y = -19$ into slope-intercept form ($y = mx + b$). First, we need to isolate the term with $y$ on one side of the equation. To do this, we'll subtract $9x$ from both sides of the equation. This gives us:

$12y = -9x - 19$

Now that we have the $y$ term isolated, we need to get $y$ by itself. Currently, $y$ is being multiplied by 12. To undo this multiplication, we'll divide both sides of the equation by 12. Make sure to divide every term on the right side by 12:

$y = \frac{-9x}{12} - \frac{19}{12}$

Now, let's simplify the fractions. The fraction $\frac{-9}{12}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us:

$y = \frac{-3x}{4} - \frac{19}{12}$

So, the equation in slope-intercept form is:

$y = -\frac{3}{4}x - \frac{19}{12}$

From this equation, we can easily identify the slope and the y-intercept. The slope, $m$, is $-\frac{3}{4}$, and the y-intercept, $b$, is $-\frac{19}{12}$. Remember, the slope tells us how steep the line is and in which direction it's going (positive or negative), while the y-intercept tells us where the line crosses the y-axis. This process is crucial for understanding and graphing linear equations. Each step ensures we maintain the equation's balance while transforming it into the desired form. Practice this with different equations to become more comfortable with the process!

Identifying Slope and Y-Intercept

Now that we've successfully converted the equation $9x + 12y = -19$ into slope-intercept form, which is $y = -\frac{3}{4}x - \frac{19}{12}$, let's pinpoint the slope and y-intercept. The slope-intercept form, as a reminder, is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. So, by comparing our transformed equation to the general form, we can easily identify these values. In our equation, $y = -\frac{3}{4}x - \frac{19}{12}$, the coefficient of $x$ is the slope. Therefore, the slope, $m$, is equal to $-\frac{3}{4}$. This means that for every 4 units we move to the right on the graph, we move 3 units down (because the slope is negative). The y-intercept is the constant term in the equation. In our case, the y-intercept, $b$, is $-\frac{19}{12}$. This tells us that the line crosses the y-axis at the point $(0, -\frac{19}{12})$. Understanding how to identify the slope and y-intercept is crucial for graphing linear equations and interpreting their properties. A negative slope indicates that the line is decreasing as you move from left to right, while a positive slope indicates an increasing line. The y-intercept provides a specific point on the y-axis that the line passes through, helping you to accurately plot the line on a coordinate plane. Grasping these concepts will greatly enhance your ability to work with linear equations and understand their graphical representations.

Graphing the Line

Having found the slope ($-\frac{3}{4}$) and the y-intercept ($-\frac{19}{12}$) from the equation $y = -\frac{3}{4}x - \frac{19}{12}$, we can now graph the line. Graphing a line using the slope-intercept form is straightforward. First, plot the y-intercept on the y-axis. In our case, the y-intercept is $-\frac{19}{12}$, which is approximately -1.58. So, find -1.58 on the y-axis and mark that point. Next, use the slope to find another point on the line. The slope is $-\frac{3}{4}$, which means for every 4 units you move to the right, you move 3 units down. Starting from the y-intercept, move 4 units to the right and 3 units down. This will give you a second point on the line. Once you have two points, simply draw a straight line through them. This line represents the equation $y = -\frac{3}{4}x - \frac{19}{12}$. When graphing, make sure to use a ruler to draw a straight line and extend it beyond the two points you've plotted. Also, it's helpful to label the line with its equation so you know which line corresponds to which equation. Graphing lines is a fundamental skill in algebra and helps you visualize the relationship between the variables in the equation. Understanding how to plot lines using the slope-intercept form makes this process much easier and more intuitive.

Common Mistakes to Avoid

When converting equations to slope-intercept form, there are a few common mistakes that students often make. One common mistake is not distributing the division correctly. When dividing both sides of the equation by a number, make sure to divide every term on both sides by that number. For example, in our equation $12y = -9x - 19$, when dividing by 12, you need to divide both $-9x$ and $-19$ by 12. Another mistake is simplifying fractions incorrectly. Always reduce fractions to their simplest form. In our example, $\frac{-9}{12}$ simplifies to $-\frac{3}{4}$. Failing to simplify fractions can lead to incorrect answers and a misunderstanding of the equation. Another mistake involves incorrectly identifying the slope and y-intercept. Remember that the slope is the coefficient of $x$, and the y-intercept is the constant term in the slope-intercept form ($y = mx + b$). Mix-ups can occur if the equation is not fully simplified or if terms are rearranged incorrectly. Finally, a common mistake is not paying attention to the signs. A negative sign can easily be dropped or added incorrectly, leading to the wrong slope or y-intercept. Always double-check your work and be careful with the signs. Avoiding these common mistakes will help you accurately convert equations to slope-intercept form and correctly interpret their properties. Always practice and double-check your work to minimize errors!

Practice Problems

To solidify your understanding of converting equations to slope-intercept form, let's go through a few practice problems. This will give you a chance to apply what you've learned and build confidence in your skills. Remember, practice is key to mastering any mathematical concept. Problem 1: Convert the equation $6x + 3y = 12$ to slope-intercept form. First, subtract $6x$ from both sides: $3y = -6x + 12$. Then, divide both sides by 3: $y = -2x + 4$. The slope is -2, and the y-intercept is 4. Problem 2: Convert the equation $4x - 8y = 24$ to slope-intercept form. Subtract $4x$ from both sides: $-8y = -4x + 24$. Then, divide both sides by -8: $y = \frac{1}{2}x - 3$. The slope is $\frac{1}{2}$, and the y-intercept is -3. Problem 3: Convert the equation $2x + 5y = -10$ to slope-intercept form. Subtract $2x$ from both sides: $5y = -2x - 10$. Then, divide both sides by 5: $y = -\frac{2}{5}x - 2$. The slope is $-\frac{2}{5}$, and the y-intercept is -2. Working through these problems will help you become more comfortable with the process. Remember to follow the steps carefully: isolate the $y$ term, divide by the coefficient of $y$, and simplify the equation. With enough practice, you'll be able to convert any linear equation to slope-intercept form with ease!

Real-World Applications

Understanding slope-intercept form isn't just about solving equations in a classroom; it has many real-world applications. Linear equations and their slope-intercept form can model various scenarios. For example, consider a taxi service that charges a fixed fee plus a per-mile rate. The fixed fee is the y-intercept, and the per-mile rate is the slope. If a taxi charges a $3 fixed fee and $2 per mile, the equation representing the total cost, $y$, for $x$ miles is $y = 2x + 3$. Another example is simple interest. If you deposit money into an account with a fixed annual interest rate, the initial deposit is the y-intercept, and the annual interest is the slope. For instance, if you deposit $100 into an account with a 5% annual interest rate, the equation representing the total amount, $y$, after $x$ years is $y = 0.05x + 100$. Slope-intercept form is also useful in physics. For example, the equation for the distance traveled at a constant speed can be written in slope-intercept form, where the speed is the slope and the initial distance is the y-intercept. Understanding these applications can help you see the relevance of linear equations in everyday life. By recognizing these patterns, you can use slope-intercept form to analyze and solve real-world problems more effectively. Whether it's calculating costs, predicting growth, or analyzing physical phenomena, the slope-intercept form provides a powerful tool for understanding linear relationships.