Slope-Intercept Form: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of linear equations and mastering the slope-intercept form. This is a fundamental concept in algebra, and understanding it will open doors to solving a variety of problems. Let's take the equation $15x - 17y = -15$ and rewrite it into the much more friendly slope-intercept form. It's like giving your equation a makeover so it's easier to understand and work with. Get ready, because by the end of this, you'll be converting equations like a pro! I'll break it down into simple, easy-to-follow steps. No complicated jargon, just clear explanations. So, grab your pencils, and let's get started. We'll be using integers, proper fractions, and improper fractions in their simplest forms. This will ensure we maintain accuracy and clarity throughout our calculations. The slope-intercept form is a cornerstone of linear equations, providing a clear understanding of a line's characteristics. This form simplifies graphing, analysis, and comparison of linear relationships. Understanding this form is essential for tackling more complex mathematical problems. So, guys, let's turn this seemingly complex equation into a straightforward one. We aim to isolate y, which allows us to identify the slope and y-intercept directly. This method ensures we can quickly grasp the essence of the linear equation. Remember, practice makes perfect. The more equations you convert, the more comfortable and confident you'll become. So, without further ado, let's begin the transformation! Think of it like a puzzle where we manipulate the equation to get y all by itself. We'll utilize basic algebraic operations to rearrange the terms. This is a critical skill in algebra, so pay close attention. It is also an important technique when you're trying to solve systems of equations. It is also important in almost every field that deals with equations. So, let's start the journey!

Step 1: Isolate the y-term

Alright, first things first, our mission is to get that y term alone on one side of the equation. To do this, we need to move the x term to the other side. Looking at our original equation, $15x - 17y = -15$, we can see that the x term is $15x$. To move it, we'll subtract $15x$ from both sides of the equation. This maintains the balance, which is super important in algebra. If we only do it on one side, we'll get a completely different equation. Doing this gives us: $15x - 17y - 15x = -15 - 15x$. When we simplify, the $15x$ terms on the left side cancel each other out, leaving us with $-17y = -15 - 15x$. Now, our equation is a little simpler, with the y term a step closer to being isolated. This initial step is fundamental. It sets the stage for isolating y completely. The key here is to perform the same operation on both sides of the equation to keep everything balanced. Remember, equations are like see-saws; to stay balanced, you must apply the same force to both sides. So, by subtracting $15x$ from both sides, we've kept our equation in equilibrium. This action ensures that the equality remains valid. Also, always remember to simplify your equation after each operation. It makes the next step easier to see and execute. Now we are one step closer to isolating y. This step is usually pretty straightforward, and with practice, you'll be doing this in your head in no time. So, take a deep breath, and let's go to the next step!

Step 2: Divide to Solve for y

Now that we've got the y term mostly isolated, our next goal is to get y completely by itself. Currently, it's being multiplied by -17. To undo that, we need to divide both sides of the equation by -17. This operation will isolate y on the left side. So, taking our equation from Step 1, which was $-17y = -15 - 15x$, we divide both sides by -17: $(-17y) / -17 = (-15 - 15x) / -17$. This simplifies to: $y = (-15 / -17) + (-15x / -17)$. Now, let's simplify further. Remember that a negative divided by a negative results in a positive. Therefore, $(-15 / -17)$ simplifies to $15/17$, and $(-15x / -17)$ simplifies to $(15/17)x$. Putting it all together, we get: $y = (15/17)x + 15/17$. Boom! We've successfully converted our equation into slope-intercept form! We have isolated y, which is the main goal. It is very important to remember that we should perform the same operation on both sides of the equation to maintain balance. This ensures the equation remains valid, and the solution stays accurate. This is really, really important in all kinds of mathematical equations, so remember to follow this rule. By dividing by -17, we've effectively undone the multiplication and isolated y. This is the core principle behind solving equations: performing inverse operations to isolate the variable. Notice how we treated each term separately when dividing. This approach ensures that we properly distribute the division across all terms on the right side of the equation. Now we can see the coefficients of the slope and the y-intercept. We're getting really close to the end, guys. Just one more step, and we'll be done!

Step 3: Identify the Slope and y-intercept

Fantastic work, everyone! Now that we have our equation in slope-intercept form, $y = (15/17)x + 15/17$, we can easily identify the slope and y-intercept. The slope-intercept form of a linear equation is written as $y = mx + b$, where m represents the slope and b represents the y-intercept. In our converted equation, $y = (15/17)x + 15/17$, the coefficient of x is $15/17$. Therefore, the slope (m) is $15/17$. The y-intercept is the constant term, which is $15/17$. So, the line crosses the y-axis at the point $(0, 15/17)$. The ability to identify the slope and y-intercept directly from the equation is one of the major benefits of using the slope-intercept form. This knowledge is invaluable for graphing the line, determining its direction (upward or downward), and understanding its relationship to the coordinate axes. It is also an important skill in real-world applications of linear equations, such as predicting future trends based on historical data. By writing the equation in slope-intercept form, we can see exactly how the line is positioned on the graph. This visual representation makes it easier to understand the equation's behavior. We can also compare and contrast the different linear equations to show the differences between them. The slope tells us how much y changes for every one-unit change in x. The y-intercept tells us where the line crosses the y-axis. Now that you have learned how to convert the form, you can find the slope and y-intercept in an instant! Great work, guys! You did it!

Final Answer

So, the slope-intercept form of the equation $15x - 17y = -15$ is $y = (15/17)x + 15/17$. The slope is $15/17$, and the y-intercept is $15/17$. Congratulations, you've successfully rewritten the equation in slope-intercept form. You've also identified the slope and y-intercept! Keep practicing, and you'll become a pro at this. Remember, the key to mastering any math concept is practice. By working through different examples and applying the steps we covered, you'll build a strong foundation and become more confident in your ability to solve linear equations. Always remember to double-check your work to ensure accuracy. If you're stuck, try working backward from your answer to see if it leads you to the original equation. Also, try to use different methods to check your answers. This will enhance your skills and reduce the likelihood of making mistakes. Also, don't be afraid to ask for help! Whether it's from a teacher, a classmate, or an online resource, seeking assistance is a sign of intelligence and a great way to learn. So, keep up the amazing work, and keep exploring the fascinating world of mathematics. The journey of learning math is not always easy, but it is super rewarding. You are all doing great, and always remember to never give up!