Solving For Y: -4x + 2y = 8 Explained Simply
Hey guys! Let's break down how to solve the equation -4x + 2y = 8 for y. This is a common type of problem in algebra, and once you get the hang of it, it's super straightforward. We'll go through each step, making sure everything is crystal clear. So, grab your pencils, and let's get started!
Understanding the Goal
The main goal here is to isolate y on one side of the equation. This means we want to rewrite the equation so it looks like y = something. This “something” will be an expression involving x and some constants. Think of it as getting y all by itself in its own little corner of the equation. When we solve for y in terms of x, we manipulate the equation to get y alone on one side. The original equation we're dealing with is -4x + 2y = 8. Our mission, should we choose to accept it, is to get this into the form y = something involving x. This means we need to get rid of everything else around the y.
Why Solve for Y?
You might be wondering, why do we even bother doing this? Well, solving for y is incredibly useful in many areas of math, especially when graphing linear equations. When an equation is in the form y = mx + b, it's super easy to graph because m represents the slope and b represents the y-intercept. This form makes it simple to visualize and understand the relationship between x and y. Plus, it's a crucial skill for more advanced math topics, so mastering it now will definitely pay off later!
Step-by-Step Solution
Okay, let's dive into the step-by-step process of solving -4x + 2y = 8 for y. We'll take it slow and steady, so you can follow along easily.
Step 1: Isolate the Term with y
First, we need to get the term with y (which is 2y) by itself on one side of the equation. To do this, we need to get rid of the -4x term. How do we do that? We use the inverse operation. Since we're subtracting 4x, we'll add 4x to both sides of the equation. Remember, whatever you do to one side, you have to do to the other to keep the equation balanced. Think of it like a scale – if you add something to one side, you need to add the same thing to the other side to keep it level.
So, we have:
-4x + 2y + 4x = 8 + 4x
The -4x and +4x on the left side cancel each other out, leaving us with:
2y = 8 + 4x
Great! We've taken the first step towards isolating y.
Step 2: Divide to Isolate y
Now we have 2y = 8 + 4x. The next step is to get y completely by itself. Right now, y is being multiplied by 2. To undo this multiplication, we need to do the inverse operation, which is division. We'll divide both sides of the equation by 2. Again, it's super important to do it to both sides to maintain the balance.
So, we have:
(2y) / 2 = (8 + 4x) / 2
On the left side, the 2s cancel out, leaving us with just y. On the right side, we need to divide both terms (8 and 4x) by 2:
y = 8/2 + (4x)/2
Now, let's simplify those fractions:
y = 4 + 2x
Step 3: Rewrite in Slope-Intercept Form (Optional, but Recommended)
We've solved for y, but it's often helpful to rewrite the equation in slope-intercept form, which is y = mx + b. This form makes it super easy to identify the slope (m) and the y-intercept (b). To do this, we just need to rearrange the terms:
y = 2x + 4
And there you have it! We've successfully solved the equation for y in terms of x.
The Answer and Why It's Correct
So, looking back at the original multiple-choice options, the correct answer is:
C. y = 4 + 2x (which is the same as y = 2x + 4)
Why?
We followed the correct algebraic steps: adding 4x to both sides and then dividing by 2. Each step was performed correctly, leading us to the isolated form of y. This matches option C, thus confirming our answer.
Common Mistakes to Avoid
Everyone makes mistakes, especially when learning something new. Here are a few common pitfalls to watch out for when solving equations for y:
- Forgetting to Distribute: When dividing both sides by a number, make sure you divide every term on that side. For example, in our problem, we divided both 8 and 4x by 2. Some people might forget to divide both, which would lead to an incorrect answer.
- Not Adding/Subtracting Correctly: Be careful with your signs! A simple mistake like adding instead of subtracting (or vice versa) can throw off your entire answer.
- Not Keeping the Equation Balanced: Remember, whatever you do to one side, you MUST do to the other. If you don't, you're changing the equation and won't get the correct solution.
- Mixing up operations: It's easy to get addition and multiplication confused, especially when you're moving terms around. Always remember to use the inverse operation: addition to undo subtraction, multiplication to undo division, and vice versa.
Practice Makes Perfect
The best way to get comfortable with solving equations for y is to practice! Here are a few similar problems you can try:
- Solve for
y: -2x + 3y = 9 - Solve for
y: 5x - y = 2 - Solve for
y: x + 4y = 12
Work through these problems step-by-step, and you'll be a pro in no time! Check your answers, and if you get stuck, go back and review the steps we covered earlier.
Real-World Applications
Solving for y isn't just a math textbook exercise. It has real-world applications too! For example, it's used in:
- Physics: Calculating the trajectory of a projectile.
- Economics: Modeling supply and demand curves.
- Computer Graphics: Creating lines and shapes on a screen.
Understanding how to manipulate equations is a powerful tool that can be used in many different fields.
Conclusion
Solving for y in terms of x might seem tricky at first, but with a little practice, it becomes second nature. Just remember the key steps: isolate the y term, divide to get y by itself, and double-check your work. And don't forget to rewrite in slope-intercept form whenever it's helpful!
Keep practicing, and you'll be solving equations like a math whiz in no time. You've got this! If you ever get stuck, remember to break the problem down step by step, and don't hesitate to ask for help. Happy solving!