Solving For U: A Step-by-Step Guide To The Equation

Hey guys! Today, we're diving into a classic algebra problem: solving for a variable. In this case, we're tackling the equation $43 = -3u - 6u - 29$ and our mission is to isolate 'u' and figure out its value. Don't worry, it's not as scary as it looks! We'll break it down step-by-step, so you can follow along and master this skill. Let's jump right in and get that 'u' figured out!

Understanding the Equation

Before we start crunching numbers, let's take a moment to really understand what this equation is telling us. We've got $43$ on one side, which is our target number. On the other side, we have a combination of terms involving 'u' and a constant. Specifically, we have $-3u$, $-6u$, and $-29$. Our goal is to manipulate this equation, using the rules of algebra, until we get 'u' all by itself on one side, revealing its value. This involves combining like terms and using inverse operations to isolate 'u'. Think of it like unwrapping a present – we need to carefully peel away the layers until we get to the core, which in this case, is the value of 'u'.

Key Concepts to Keep in Mind:

• Variables: 'u' is our variable, the unknown value we're trying to find.
• Constants: $43$ and $-29$ are constants, fixed numerical values.
• Coefficients: $-3$ and $-6$ are coefficients, the numbers multiplying the variable 'u'.
• Terms: $-3u$, $-6u$, and $-29$ are individual terms in the equation.
• Inverse Operations: We'll use inverse operations (addition/subtraction, multiplication/division) to isolate 'u'. This is a fundamental principle in solving equations. Whatever operation we perform on one side of the equation, we must perform the same operation on the other side to maintain the balance. For instance, if we add a number to one side, we must add the same number to the other side.

Understanding these basic concepts is crucial for tackling any algebraic equation. Now that we've got a solid grasp of the equation's anatomy, let's move on to the first step in solving for 'u'.

Step 1: Combine Like Terms

The first thing we want to do to simplify our equation $43 = -3u - 6u - 29$ is to combine the like terms. Remember, like terms are those that have the same variable raised to the same power. In this case, we have $-3u$ and $-6u$. These both have 'u' to the power of 1, so we can combine them. Think of it like having -3 apples and then getting -6 more apples. How many apples do you have in total? You'd have -9 apples, right? The same logic applies here.

To combine these terms, we simply add their coefficients: $-3 + (-6) = -9$. So, $-3u - 6u$ becomes $-9u$. Now our equation looks like this:

$43 = -9u - 29$

See how much simpler that is already? Combining like terms is a crucial step in simplifying equations and making them easier to solve. It reduces the clutter and allows us to focus on isolating the variable we're interested in. By grouping together terms that share the same variable, we streamline the equation and pave the way for the next steps in the solution process. This simplification technique is not just limited to this specific equation; it's a widely used strategy in algebra for dealing with expressions and equations of varying complexity.

Now that we've combined the 'u' terms, we're one step closer to isolating 'u'. Let's move on to the next step and see how we can further simplify the equation to get 'u' all by itself.

Step 2: Isolate the Term with 'u'

Okay, we've simplified our equation to $43 = -9u - 29$. Now, we need to isolate the term with 'u', which is $-9u$. This means we want to get $-9u$ by itself on one side of the equation. To do this, we need to get rid of the $-29$ that's hanging out on the same side. Remember how we talked about inverse operations? That's exactly what we're going to use here.

The opposite of subtracting 29 is adding 29. So, to get rid of the $-29$, we'll add 29 to both sides of the equation. This is super important! Whatever we do to one side, we have to do to the other to keep the equation balanced. It's like a see-saw – if you add weight to one side, you need to add the same weight to the other to keep it level.

So, let's add 29 to both sides:

$43 + 29 = -9u - 29 + 29$

On the left side, $43 + 29 = 72$. On the right side, $-29 + 29$ cancels out, leaving us with just $-9u$. So, our equation now looks like this:

$72 = -9u$

We're getting closer! We've successfully isolated the term with 'u'. This step demonstrates a fundamental principle in solving equations: using inverse operations to systematically eliminate terms and bring us closer to our goal of isolating the variable. By adding 29 to both sides, we effectively neutralized the $-29$ on the right side, allowing us to focus solely on the term containing 'u'. This strategic manipulation is a key technique in algebra, enabling us to unravel the complexities of equations and reveal the values of unknown variables.

Step 3: Solve for 'u'

Alright, we've reached the final step! We've got our equation down to $72 = -9u$. Now, we just need to get 'u' completely by itself. Right now, 'u' is being multiplied by $-9$. So, to undo this multiplication, we need to use the inverse operation: division.

We're going to divide both sides of the equation by $-9$. Remember, whatever we do to one side, we have to do to the other to keep things balanced. So, let's divide:

rac{72}{-9} = rac{-9u}{-9}

On the left side, $72$ divided by $-9$ is $-8$. On the right side, the $-9$ in the numerator and the $-9$ in the denominator cancel each other out, leaving us with just 'u'. So, our equation becomes:

$-8 = u$

And there you have it! We've solved for 'u'! Our solution is $u = -8$. This final step highlights the power of inverse operations in isolating variables. By dividing both sides of the equation by -9, we effectively undid the multiplication that was binding 'u', allowing us to reveal its true value. This principle of using inverse operations is a cornerstone of algebraic problem-solving, applicable across a wide range of equations and scenarios. This technique allows us to systematically peel away the layers surrounding the variable until we arrive at the solution.

Step 4: Check Your Solution

Now, before we celebrate too much, it's always a good idea to check our answer. This is a super important step because it helps us catch any mistakes we might have made along the way. To check our solution, we're going to plug $u = -8$ back into the original equation:

$43 = -3u - 6u - 29$

Substitute $u$ with $-8$:

$43 = -3(-8) - 6(-8) - 29$

Now, let's simplify. Remember the order of operations (PEMDAS/BODMAS)? Multiplication comes before addition and subtraction.

$43 = 24 + 48 - 29$

Now, let's add and subtract:

$43 = 72 - 29$

$43 = 43$

Woohoo! The left side of the equation equals the right side of the equation. This means our solution, $u = -8$, is correct! Checking our solution is not just a formality; it's a crucial step in ensuring accuracy. By plugging our solution back into the original equation, we verify that our answer satisfies the equation's conditions. This process reinforces our understanding of the equation and its solution, solidifying our confidence in the result. It's a habit that all successful problem-solvers adopt to minimize errors and maximize their understanding.

Conclusion

Awesome job, guys! We successfully solved the equation $43 = -3u - 6u - 29$ for 'u', and we found that $u = -8$. We did this by following a step-by-step process:

1. Combined like terms.
2. Isolated the term with 'u'.
3. Solved for 'u' using inverse operations.
4. Checked our solution to make sure it was correct.

Solving algebraic equations is a fundamental skill in mathematics, and mastering these steps will set you up for success in more advanced topics. Remember, the key is to break down complex problems into smaller, manageable steps. By understanding the underlying concepts and applying them systematically, you can tackle even the trickiest equations. Keep practicing, and you'll become a pro at solving for any variable! This process not only helps us find the solution but also strengthens our understanding of algebraic principles and problem-solving strategies. So, keep practicing, keep exploring, and keep challenging yourself with new equations. The more you practice, the more confident and proficient you'll become in your mathematical abilities.