Solve For Y: Y=2x+2 When X=-3

by Sebastian Müller 30 views

Hey guys! Today, we're diving into a super common algebra problem: solving for y when we know the value of x in a linear equation. Specifically, we're going to tackle the equation y = 2x + 2, and we're given that x = -3. Don't worry, it's not as scary as it sounds! We'll break it down step-by-step so that everyone can follow along. Understanding how to solve these types of problems is crucial because they form the foundation for more advanced math concepts. These skills aren't just for the classroom; they pop up in real-world situations all the time, from calculating costs to understanding graphs and charts. So, let's jump right in and make sure we've got this down pat!

Understanding the Equation

Before we start plugging in numbers, let's make sure we understand what the equation y = 2x + 2 is telling us. This is a linear equation, which means that when we graph it, we'll get a straight line. The equation itself shows the relationship between two variables, x and y. The value of y depends on the value of x, which makes x the independent variable and y the dependent variable. The equation is in slope-intercept form (y = mx + b), where 'm' represents the slope of the line and 'b' represents the y-intercept. The slope (m) tells us how much y changes for every one unit change in x. In our case, the slope is 2, which means that for every increase of 1 in x, y increases by 2. The y-intercept (b) is the point where the line crosses the y-axis. In our equation, the y-intercept is 2, meaning the line crosses the y-axis at the point (0, 2). Grasping these concepts is key to not only solving this particular problem but also understanding linear equations in general. Linear equations are everywhere – from simple budgeting to complex scientific models – so getting comfortable with them now will pay off big time later. Remember, math is like building blocks; each concept builds on the previous one. So, let's solidify our understanding of linear equations and move on to the next step: substituting the value of x.

Substituting x = -3

Okay, now that we've got a good grasp of what our equation means, let's get down to the nitty-gritty of solving it. We know that x = -3, and our mission is to find the corresponding value of y. The magic here is substitution. We're going to replace the x in our equation with the number -3. So, our equation y = 2x + 2 becomes y = 2(-3) + 2. See how we just swapped out the x for -3? It's like replacing a piece in a puzzle. Now, it's super important to remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order is our roadmap for solving the equation correctly. If we don't follow it, we might end up with the wrong answer, and nobody wants that! So, let's keep PEMDAS/BODMAS in the back of our minds as we move on to the next step: simplifying the equation. By carefully substituting and keeping the order of operations in check, we're setting ourselves up for success. Remember, practice makes perfect, so the more you work through these types of problems, the more comfortable you'll become with the process. Substitution is a fundamental skill in algebra, and it's used in countless other mathematical contexts. So, mastering it here will definitely give you a leg up in your math journey.

Simplifying the Equation

Alright, we've substituted x = -3 into our equation, and now we have y = 2(-3) + 2. The next step is to simplify this expression and get y all by itself on one side of the equation. Remember PEMDAS/BODMAS? It's time to put it into action! According to the order of operations, we need to handle multiplication before we tackle the addition. So, let's multiply 2 by -3. 2 multiplied by -3 equals -6. This is a crucial step, and it's super important to get the sign right. A positive number times a negative number always results in a negative number. Now our equation looks like this: y = -6 + 2. We've taken care of the multiplication, so now we move on to addition. We're adding 2 to -6. Think of it like starting at -6 on a number line and moving 2 steps to the right. Where do we end up? We end up at -4. So, -6 + 2 = -4. This means our simplified equation is y = -4. We've done it! We've successfully simplified the equation and found the value of y when x = -3. This process of simplifying expressions using the order of operations is a cornerstone of algebra, and it's a skill you'll use over and over again. By carefully following PEMDAS/BODMAS, we can break down even complex equations into manageable steps. Remember, each step is important, and double-checking your work can help you avoid those little mistakes that can sometimes trip us up. Now that we've found the value of y, let's make sure we understand what our answer means in the context of the original problem.

The Solution

Okay, fantastic! We've gone through the steps, substituted, simplified, and arrived at our solution: y = -4. But what does this actually mean? Well, in the context of our original equation, y = 2x + 2, when x is -3, the corresponding value of y is -4. We can express this as an ordered pair: (-3, -4). This ordered pair represents a point on the line that our equation describes. Remember, linear equations represent straight lines, and every point on that line satisfies the equation. So, the point (-3, -4) sits right there on the line y = 2x + 2. To further solidify our understanding, we could even graph this line. If we plotted the point (-3, -4) and another point on the line (like the y-intercept, which we know is (0, 2)), we could draw a line through them and visualize the solution. This connection between algebra and graphing is super powerful, and it's something you'll explore in more depth as you continue your math journey. Understanding the solution not just as a number, but as a point on a line, gives you a more complete picture of what's going on. It's like seeing the whole puzzle instead of just one piece. So, let's take a moment to appreciate what we've accomplished. We've successfully solved for y in a linear equation, and we've connected our solution to the concept of graphing. Great job, guys! Now, let's wrap things up with a quick recap of what we've learned.

Recap

Alright, let's take a step back and recap everything we've covered in this problem. We started with the equation y = 2x + 2 and the information that x = -3. Our goal was to find the corresponding value of y. We tackled this problem using a straightforward, step-by-step approach. First, we understood the equation, recognizing it as a linear equation in slope-intercept form. We identified the slope and y-intercept, which gave us a good foundation for understanding the relationship between x and y. Next, we substituted the value of x (-3) into the equation, replacing the x with -3. This gave us a new equation: y = 2(-3) + 2. Then, we simplified the equation using the order of operations (PEMDAS/BODMAS). We multiplied 2 by -3 to get -6, and then we added 2 to -6, which gave us -4. Finally, we arrived at our solution: y = -4. We also interpreted this solution as an ordered pair, (-3, -4), which represents a point on the line described by the equation. By breaking the problem down into these clear steps, we made it much more manageable. This approach – understanding, substituting, simplifying, and interpreting – is a powerful tool for solving all sorts of math problems. Remember, math is often about taking complex problems and breaking them down into smaller, more digestible pieces. So, keep practicing these steps, and you'll be well on your way to mastering algebra! Now that we've recapped, you've got a solid understanding of how to solve this type of problem. Keep up the great work, and don't hesitate to tackle more challenges!