Find Output Y When X = -9: Step-by-Step Guide

Hey everyone! Today, we're going to tackle a super common type of math problem: finding the output of a function when you're given a specific input. In simpler terms, we're going to figure out what y is when x is -9. This is a fundamental concept in algebra and is used all the time in various applications, from graphing equations to understanding relationships between variables. So, let's dive in and break it down step-by-step!

Understanding the Basics

Before we jump into solving the problem, let's make sure we're all on the same page with the basics. In mathematics, a function is like a machine: you put something in (the input, often represented by x), and the machine spits something out (the output, often represented by y). The function tells you exactly what to do with the input to get the output. This relationship is often written as an equation, like y = f(x), where f is the function. So, when we say "find the output y when the input x is -9," we're essentially asking, "If we plug -9 into the function, what value will we get for y?"

To really grasp this, think of it like a recipe. The input x is like the ingredients you have, the function is the recipe itself (the instructions on what to do with the ingredients), and the output y is the final dish. If you change the ingredients (the input), you'll likely change the final dish (the output). This analogy helps visualize how changing x affects y. This is crucial because many real-world scenarios can be modeled using functions. For instance, the distance a car travels depends on its speed and the time it travels. Here, speed and time are inputs, and distance is the output. Similarly, the cost of a product can be a function of the materials used and the labor involved. Understanding how to find the output for a given input is therefore essential for solving practical problems.

Furthermore, visualizing functions graphically can provide a deeper understanding. When we plot the relationship between x and y on a graph, we create a visual representation of the function. Each point on the graph corresponds to a pair of (x, y) values. The x-coordinate represents the input, and the y-coordinate represents the corresponding output. For example, if the point (-9, 5) lies on the graph of a function, it means that when the input x is -9, the output y is 5. This graphical representation can help us quickly identify the output for any given input, or vice versa. This is particularly useful for functions that are difficult to express algebraically or when we need to understand the overall behavior of the function. So, understanding the concept of inputs and outputs is not just about solving equations; it's about understanding relationships and patterns, which are fundamental to many areas of mathematics and science.

The Problem: Finding y When x = -9

Now, let's get down to the specific problem. We need the equation that relates x and y. Without the equation, we're stuck! Let's assume, for the sake of demonstration, that the equation is: y = 2x + 5. Okay, so we have a simple linear equation here. This equation is our rule, our recipe, our machine. It tells us exactly what to do with the input x to get the output y. Remember, our goal is to find the value of y when x is -9.

The beauty of algebra is its systematic approach. Once we have the equation, solving for y is a straightforward process. We simply substitute the given value of x into the equation and perform the arithmetic. This is where the concept of substitution becomes crucial. Substitution is a fundamental technique in mathematics where we replace a variable with its corresponding value. In this case, we are substituting x with -9. This might seem simple, but it's a powerful tool that allows us to solve a wide range of problems. It's important to be careful with signs and order of operations when substituting values, as errors in these steps can lead to incorrect results. For example, if we have a negative value for x, we need to make sure to include the negative sign when we substitute it into the equation. Similarly, we need to follow the order of operations (PEMDAS/BODMAS) to ensure that we perform the calculations in the correct sequence. This ensures that we arrive at the correct solution.

Understanding the equation is also key to interpreting the result. The equation y = 2x + 5 tells us that y is equal to two times x, plus 5. This means that for every increase of 1 in x, y will increase by 2. The constant term, 5, represents the y-intercept, which is the value of y when x is 0. Understanding these relationships can help us predict the behavior of the function and interpret the results in a meaningful way. For instance, if we were modeling the cost of a service as a function of the number of hours worked, the equation might be y = 20x + 50, where y is the total cost, x is the number of hours worked, 20 is the hourly rate, and 50 is a fixed fee. In this case, we can see that the cost increases by $20 for each additional hour of work, and the initial cost is $50, regardless of the number of hours worked.

Solving for y

Alright, let's do this! We have y = 2x + 5 and x = -9. The next step is very important: we substitute -9 for x in the equation. This means we replace the x in the equation with -9. It's like taking the x out and putting -9 in its place. So, our equation becomes y = 2(-9) + 5. See how we just swapped x for -9? Make sure you use parentheses, especially when dealing with negative numbers. This helps avoid confusion and ensures you multiply correctly.

The next step involves following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). These acronyms serve as a roadmap for solving mathematical expressions. In our case, we first need to perform the multiplication before we do the addition. This means we multiply 2 by -9, which gives us -18. So, our equation now looks like this: y = -18 + 5. This is a much simpler equation to solve. It's crucial to follow the order of operations because changing the order can lead to a completely different answer. For example, if we were to add 5 to -9 first, and then multiply by 2, we would get a different result. So, following PEMDAS/BODMAS is essential for accurate calculations.

Now, we're in the home stretch! We have y = -18 + 5. This is a simple addition problem involving a negative and a positive number. Remember the rules for adding integers: when adding numbers with different signs, we subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value. In this case, the absolute value of -18 is 18, and the absolute value of 5 is 5. Subtracting 5 from 18 gives us 13. Since -18 has a larger absolute value and is negative, our final answer will be negative. Therefore, -18 + 5 equals -13. So, finally, we have our answer!

The Answer

So, guys, when x = -9, y = -13. That's it! We found the output. We took the input (-9), plugged it into our function (y = 2x + 5), and calculated the output (-13). This is a fundamental skill in algebra and one you'll use constantly. You can think of it like this: the function is a rule, and we're just following the rule to get the answer. This process of substitution and simplification is key to solving many mathematical problems. The equation gives us a relationship between x and y, and by knowing one variable, we can find the other. This is not just applicable in mathematics but also in various real-world scenarios where relationships between quantities are expressed as functions. For example, if you know the speed of a car and the time it has traveled, you can use a function to calculate the distance it has covered.

To solidify your understanding, try practicing with different functions and inputs. You can change the equation (for example, try y = -3x + 7 or y = x^2 - 4) and plug in different values for x. The more you practice, the more comfortable you'll become with this process. It's also helpful to visualize the function graphically. By plotting the points (x, y) on a graph, you can see how the output y changes as the input x varies. This visual representation can give you a deeper understanding of the function and its behavior. Remember, mathematics is not just about memorizing formulas; it's about understanding concepts and developing problem-solving skills. So, keep practicing, keep asking questions, and you'll become more confident in your mathematical abilities.

Practice Makes Perfect

To really nail this down, try some practice problems! Change the equation, change the value of x, and work through the steps. The more you practice, the easier it will become. For example, what if the equation was y = -x^2 + 3 and x = 2? Or what if the equation was y = (x + 1) / 2 and x = 5? Working through different examples will help you see how the process works for different types of equations. Don't be afraid to make mistakes – mistakes are a great way to learn! When you encounter a mistake, take the time to understand why you made it and how to correct it. This will help you avoid making the same mistake in the future. Also, don't hesitate to seek help if you're struggling. Ask your teacher, classmates, or look for online resources. There are many websites and videos that can provide additional explanations and examples. Learning mathematics is a journey, and it's okay to ask for help along the way.

Remember, guys, math is like a muscle – the more you exercise it, the stronger it gets! So keep practicing, and you'll be finding outputs like a pro in no time!

If you're still having trouble, break the problem down into smaller steps. Sometimes, the problem seems daunting because it's presented as one big task. But if you break it down into smaller, more manageable steps, it becomes much easier to handle. For example, instead of trying to solve the entire equation at once, focus on each operation individually. First, substitute the value of x into the equation. Then, perform any multiplications or divisions. Finally, perform any additions or subtractions. This step-by-step approach can make the problem less overwhelming and help you avoid mistakes. Also, consider drawing a diagram or using a visual aid to help you understand the problem. Sometimes, visualizing the problem can make it easier to solve. For example, you can plot the function on a graph and see how the output changes as the input varies. This can give you a better understanding of the relationship between x and y and help you solve the problem more effectively.