Hey guys! Let's dive into a super common type of math problem you'll often see in algebra: solving for a variable. In this case, we're going to tackle the equation 2x - 3 = y - 4x and figure out the value of x. Don't worry, it might look a little intimidating at first, but we'll break it down step by step so it's totally manageable.
Understanding the Goal: Isolating X
The heart of solving for a variable, like x in this equation, is to isolate it. What does that mean? It simply means we want to manipulate the equation using mathematical operations until we have x all by itself on one side of the equals sign. Think of it like building a fortress around x, carefully moving everything else away until it stands alone in its glorious, solved state. To do this, we'll use the fundamental principles of algebra, which allow us to perform the same operations on both sides of the equation without changing its balance. Imagine it as a scale – if you add or subtract something from one side, you need to do the same on the other to keep it level. This ensures that the equation remains true throughout our solving process. Our roadmap involves strategic additions, subtractions, multiplications, or divisions, all with the ultimate aim of getting x by its lonesome. So, let's roll up our sleeves and get started on this algebraic adventure!
Step 1: Gathering the X Terms
Our initial equation is 2x - 3 = y - 4x. The first order of business is to gather all the terms containing x on the same side of the equation. Looking at our equation, we have 2x on the left side and -4x on the right side. To bring the -4x over to the left, we need to get rid of it from the right side. The way we do this is by performing the inverse operation. Since we have a subtraction of 4x, the inverse operation is addition. So, we're going to add 4x to both sides of the equation. This is a crucial step in isolating x, as it begins the process of combining like terms and simplifying the equation. Remember, whatever we do to one side, we must do to the other to maintain the balance and ensure the equation remains valid. This principle is the cornerstone of algebraic manipulation, allowing us to rearrange and simplify equations while preserving their fundamental truth. By adding 4x to both sides, we are one step closer to our goal of having all the x terms together, making it easier to isolate x and solve for its value. Let's write it out:
2x - 3 + 4x = y - 4x + 4x
Now, let's simplify both sides.
Step 2: Simplifying the Equation
After adding 4x to both sides, our equation looks like this: 2x - 3 + 4x = y - 4x + 4x. Now it's time to simplify! On the left side, we have two terms with x: 2x and 4x. We can combine these like terms by simply adding their coefficients (the numbers in front of the x). So, 2 + 4 equals 6, giving us 6x. The left side now becomes 6x - 3. On the right side, we have -4x and +4x. These are additive inverses, meaning they cancel each other out, leaving us with just y. So, the right side simplifies to y. Our equation has now been transformed into the much simpler form: 6x - 3 = y. This simplification is a critical step in solving for x, as it reduces the complexity of the equation and brings us closer to isolating the variable. By combining like terms and eliminating inverses, we've effectively cleared away some of the clutter, making it easier to see the path forward to our solution. This step highlights the power of algebraic manipulation in making complex problems more manageable. With our simplified equation, we're now in a better position to continue isolating x and ultimately find its value.
Step 3: Moving the Constant Term
We're making great progress! Our equation is currently 6x - 3 = y. Remember, our goal is to get x all by itself. Right now, we have a -3 on the same side as the x term. To get rid of it, we need to perform the inverse operation. Since it's being subtracted, we'll add 3 to both sides of the equation. This is another crucial step in isolating x, as it moves us closer to having only the x term on one side. By adding 3 to both sides, we maintain the balance of the equation, ensuring that our manipulations are valid and lead us to the correct solution. This step demonstrates the strategic use of inverse operations to isolate variables, a fundamental technique in algebra. Let's see how it looks:
6x - 3 + 3 = y + 3
Now, let's simplify again.
Step 4: Simplifying Again
After adding 3 to both sides, we have 6x - 3 + 3 = y + 3. Let's simplify this. On the left side, we have -3 and +3, which, just like before, are additive inverses and cancel each other out. This leaves us with just 6x on the left side. The right side is simply y + 3, as these terms cannot be combined because they are not like terms. So, our equation now looks like this: 6x = y + 3. We're getting so close to having x isolated! This simplification is a key step in our journey, as it eliminates the constant term from the side with x, bringing us closer to our goal. By strategically using inverse operations and simplifying, we're unraveling the equation, making it easier to see the value of x. With each step, we're transforming the equation into a more manageable form, highlighting the power of algebraic techniques in problem-solving. Now that we have 6x = y + 3, we're just one step away from the finish line!
Step 5: Isolating X Completely
We've reached the final step! Our equation is now 6x = y + 3. The last thing standing between x and complete isolation is the coefficient 6. Remember, 6x means 6 times x. To undo this multiplication, we need to perform the inverse operation, which is division. So, we'll divide both sides of the equation by 6. This is the final piece of the puzzle, the step that will reveal the value of x in terms of y. By dividing both sides by 6, we maintain the balance of the equation while effectively isolating x. This step underscores the importance of understanding inverse operations in algebra, as they allow us to systematically unravel equations and solve for unknown variables. Let's perform the division:
(6x) / 6 = (y + 3) / 6
Now, let's simplify one last time.
Step 6: The Grand Finale – The Value of X
After dividing both sides by 6, we have (6x) / 6 = (y + 3) / 6. On the left side, the 6 in the numerator and the 6 in the denominator cancel each other out, leaving us with just x. On the right side, we have (y + 3) / 6, which we can leave as it is or rewrite as y/6 + 3/6, which simplifies to y/6 + 1/2. This is the value of x! We've successfully isolated x and expressed it in terms of y. Our final answer is:
x = (y + 3) / 6 or x = y/6 + 1/2
And there you have it, guys! We've cracked the code and found the value of x in the equation 2x - 3 = y - 4x. It might have seemed tricky at first, but by breaking it down step by step, using inverse operations, and simplifying, we were able to solve it. Remember, the key to algebra is to take your time, stay organized, and don't be afraid to tackle those equations head-on. You got this!
Key Takeaways
- Isolate the Variable: The core strategy in solving for a variable is to get it by itself on one side of the equation.
- Inverse Operations are Your Friends: Use inverse operations (addition/subtraction, multiplication/division) to move terms around.
- Keep the Balance: Whatever you do to one side of the equation, you must do to the other.
- Simplify, Simplify, Simplify: Combining like terms and canceling out opposites makes the equation easier to work with.
Practice Makes Perfect
Now that we've solved this one together, try tackling similar equations on your own. The more you practice, the more comfortable you'll become with these algebraic techniques. Remember, math is like any other skill – it gets easier with practice!