Solving Equations A Step-by-Step Guide With Karissa's Example

Hey guys! Ever get stuck in the middle of an equation and feel like you're going in circles? Well, we're going to break down a problem where Karissa tackles an equation, and we'll follow her steps to see how she nails it. It’s like we’re her study buddies, cheering her on and learning along the way. So, let's dive into this mathematical adventure!

Unraveling the Initial Equation

Karissa starts with the equation:

1/2(x - 14) + 11 = 1/2x - (x - 4)

Okay, so the first thing we notice is that we've got fractions and parentheses. Don't let that scare you! The key here is to take it one step at a time. We need to simplify both sides of the equation before we can really start solving for 'x'. Think of it like untangling a knot – you don't just yank it; you gently work through each twist and turn. So, let’s look at what Karissa did next.

Distributive Property: Karissa's First Move

The next line in Karissa's work is:

1/2x - 7 + 11 = 1/2x - x + 4

Ah, she used the distributive property! Remember that? It's when you multiply the term outside the parentheses by each term inside. On the left side, she distributed the 1/2 across (x - 14), which gives us (1/2) * x = 1/2x and (1/2) * -14 = -7. Makes sense, right? Now, on the right side, she distributed the negative sign (which is like multiplying by -1) across (x - 4). So, -1 * x = -x and -1 * -4 = +4. See how those parentheses just melted away? This is a crucial step in simplifying the equation and making it less intimidating. Distributive property is your friend, guys!

Combining Like Terms: Making It Simpler

Now, let's look at the next step. Karissa transformed the equation into:

1/2x + 4 = -1/2x + 4

Here, she combined the like terms on each side. On the left, she had -7 + 11, which simplifies to +4. Easy peasy! On the right side, she combined 1/2x - x. Now, remember that -x is the same as -1x, so she essentially did 1/2x - 1x. To subtract these, you need a common denominator, so think of 1x as 2/2x. Then, 1/2x - 2/2x = -1/2x. The equation is really starting to shape up now. It’s like the fog is clearing, and we can see the path ahead. By combining like terms, Karissa has made the equation much easier to handle. This is a technique that you will use all the time in algebra, so it is a good idea to get comfortable with it now.

Isolating the Variable: Karissa's Goal

Our main goal in solving any equation is to isolate the variable – in this case, 'x'. That means getting 'x' all by itself on one side of the equation. Karissa needs to maneuver the terms around until she achieves this. Think of it like a puzzle – each move needs to bring us closer to the final solution. Let's see how Karissa does it!

Moving 'x' Terms: Getting Closer to the Solution

So, let's say Karissa wants to move all the 'x' terms to the left side. To do that, she needs to get rid of the -1/2x term on the right side. The way to do that is to add 1/2x 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. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it level.

If she adds 1/2x to both sides, the equation becomes:

1/2x + 1/2x + 4 = -1/2x + 1/2x + 4

Which simplifies to:

x + 4 = 4

Notice how -1/2x + 1/2x cancels out on the right side, leaving just the 4. We're getting closer! The 'x' terms are now all on the left side, which is exactly what we wanted. This is a common strategy in solving equations, and it’s all about using inverse operations to move terms around.

Isolating 'x': The Final Step

Now, we just have one more step to isolate 'x'. We have x + 4 = 4. To get 'x' by itself, we need to get rid of that +4 on the left side. The opposite of adding 4 is subtracting 4, so we subtract 4 from both sides:

x + 4 - 4 = 4 - 4

This simplifies to:

x = 0

Boom! Karissa has solved for 'x'. The solution is x = 0. It's like reaching the summit of a mountain – after all that climbing, you finally get to see the amazing view. In this case, the amazing view is the solution to the equation. Karissa’s final answer is x=0. This demonstrates the power of using inverse operations to isolate the variable and solve the equation. Great job, Karissa!

Checking the Solution: Ensuring Accuracy

But hold on a second! We're not quite done yet. It's always a good idea to check your solution to make sure it's correct. This is like proofreading an essay or double-checking your work – it helps you catch any mistakes and ensures that your answer is accurate. So, how do we check our solution?

Plugging It Back In: The Verification Process

To check if x = 0 is the correct solution, we plug it back into the original equation:

1/2(x - 14) + 11 = 1/2x - (x - 4)

Replace every 'x' with '0':

1/2(0 - 14) + 11 = 1/2(0) - (0 - 4)

Now, simplify each side:

Left side:

1/2(-14) + 11 = -7 + 11 = 4

Right side:

1/2(0) - (-4) = 0 + 4 = 4

Both sides equal 4! That means our solution is correct. Woo-hoo! Checking your solution is such an important part of the problem-solving process. It’s like having a safety net – it gives you confidence that your answer is right. By plugging the solution back into the original equation, Karissa has verified that her answer is correct. This is a fantastic habit to develop, and it will help you avoid making careless mistakes.

Common Pitfalls and How to Avoid Them

Solving equations can be tricky, and there are a few common mistakes that people often make. Let's talk about some of these pitfalls and how to avoid them. Think of it as learning from other people's experiences – we can avoid the same mistakes by being aware of them.

Distribution Errors: A Tricky Trap

One common mistake is messing up the distributive property. Remember, you need to multiply the term outside the parentheses by every term inside. For example, if you have 2(x + 3), you need to multiply the 2 by both the 'x' and the '3'. So, it should be 2x + 6, not just 2x + 3. Always double-check that you've distributed correctly. It’s so easy to make a little slip-up here, so pay close attention to the signs and the numbers.

Sign Errors: A Silent Menace

Another frequent error is with signs – especially when distributing a negative sign. For instance, if you have -(x - 2), you need to distribute the negative sign to both terms inside the parentheses. That means -(x - 2) becomes -x + 2, not -x - 2. Keep a close eye on those negative signs! They can be sneaky little devils! Always take your time and be careful with negative signs. They can easily trip you up if you're not paying attention.

Combining Unlike Terms: A Big No-No

One more common mistake is trying to combine terms that aren't like terms. You can only combine terms that have the same variable raised to the same power. For example, you can combine 3x and 5x to get 8x, but you can't combine 3x and 5x^2. They're different! Make sure you're only combining like terms. It’s like trying to add apples and oranges – they’re both fruit, but they’re not the same thing. Stick to combining terms that are truly alike. This will prevent a lot of confusion and errors.

Alternative Approaches to Solving Equations

While Karissa used a standard approach to solve the equation, there are often multiple ways to tackle a problem. Let's explore some alternative methods for solving equations. It’s like having different tools in your toolbox – sometimes one tool works better than another, depending on the job.

Clearing Fractions: A Different Perspective

Instead of distributing the 1/2 at the beginning, another approach is to clear the fractions right away. To do this, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the only denominator is 2, so we multiply both sides by 2:

2 * [1/2(x - 14) + 11] = 2 * [1/2x - (x - 4)]

Distribute the 2 on both sides:

(x - 14) + 22 = x - 2(x - 4)

Notice how the fractions are gone! Now, the equation looks a bit simpler to work with. This method can be especially helpful when dealing with equations that have multiple fractions. Clearing fractions can often make an equation easier to handle, especially when you're dealing with multiple fractions. It's like getting rid of the clutter so you can focus on the core of the problem.

Different Orders of Operation: Flexibility is Key

Sometimes, you can solve an equation in different orders and still get the same answer. For example, instead of distributing the 1/2 on the left side first, Karissa could have added 11 to both sides first. The order of operations can be flexible, as long as you follow the correct rules. Being flexible with your approach can help you find the most efficient way to solve a problem. It's like finding the best route to your destination – sometimes the scenic route is just as good as the highway.

Conclusion: Mastering the Art of Equation Solving

So, there you have it! We've followed Karissa's journey step-by-step as she solved the equation 1/2(x - 14) + 11 = 1/2x - (x - 4). We saw how she used the distributive property, combined like terms, and isolated the variable to find the solution, x = 0. We also talked about the importance of checking your solution and common pitfalls to avoid. Equation solving is a fundamental skill in math, and with practice, you can master it. Remember, it’s like learning any new skill – it takes time, effort, and a willingness to learn from your mistakes. Keep practicing, and you'll become an equation-solving pro in no time! You got this, guys!