Solving Linear Equations Step-by-Step Example (1/4)x + 4 + (3/4)x - 2 = 7

Hey guys! Today, we're going to break down a math problem step-by-step. We'll be tackling an equation with fractions and variables, making sure everything is super clear so you can follow along easily. This equation involves solving for 'x', and we'll use some fundamental algebraic techniques to get there. So, grab your pencils and let's dive in!

The Equation at Hand

Our mission, should we choose to accept it (and we do!), is to solve the following equation:

$$\frac{1}{4}x + 4 + \frac{3}{4}x - 2 = 7$$

This might look a little intimidating at first glance with the fractions and all, but don't worry! We'll take it one step at a time. Our main goal here is to isolate 'x' on one side of the equation. This means we need to gather all the 'x' terms together, combine the constants, and then do some simple operations to finally figure out what 'x' equals.

Combining Like Terms

The first thing we want to do is simplify the equation by combining the terms that are similar. In this case, we have two terms with 'x' in them: $\frac{1}{4}x$ and $\frac{3}{4}x$. We also have some constant terms: 4 and -2. Let’s group these together to make things clearer.

So, we rewrite the equation as:

$$\left(\frac{1}{4}x + \frac{3}{4}x\right) + (4 - 2) = 7$$

Now, let's add the 'x' terms. Since they have the same denominator (4), it’s pretty straightforward:

$$\frac{1}{4}x + \frac{3}{4}x = \frac{1+3}{4}x = \frac{4}{4}x = 1x = x$$

Next, let’s combine the constants:

$$4 - 2 = 2$$

So, our equation now looks much simpler:

$$x + 2 = 7$$

Isolating x

Great! We're making progress. Now, to get 'x' all by itself on one side of the equation, we need to get rid of the +2. We do this by subtracting 2 from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced.

$$x + 2 - 2 = 7 - 2$$

This simplifies to:

$$x = 5$$

And there we have it! We've solved for 'x'.

Verifying the Solution

It's always a good idea to double-check our work to make sure we didn't make any mistakes. To do this, we'll plug our solution, x = 5, back into the original equation and see if it holds true.

Our original equation was:

$$\frac{1}{4}x + 4 + \frac{3}{4}x - 2 = 7$$

Substitute x = 5:

$$\frac{1}{4}(5) + 4 + \frac{3}{4}(5) - 2$$

Now, let’s simplify step by step:

$$\frac{5}{4} + 4 + \frac{15}{4} - 2$$

First, let’s combine the fractions:

$$\frac{5}{4} + \frac{15}{4} = \frac{5 + 15}{4} = \frac{20}{4} = 5$$

Now, our equation looks like this:

$$5 + 4 - 2$$

Next, let’s add and subtract the constants:

$$5 + 4 = 9$$

$$9 - 2 = 7$$

So, we have:

$$7 = 7$$

Since both sides of the equation are equal, our solution x = 5 is correct! We nailed it!

The Final Answer

After carefully solving the equation and verifying our answer, we can confidently say that:

$$x = 5$$

So, the correct answer is:

D. 5

Recap

Let’s quickly recap the steps we took to solve this equation:

1. Combined Like Terms: We grouped the 'x' terms together and the constant terms together.
2. Simplified the Equation: We added the fractions and combined the constants.
3. Isolated x: We subtracted 2 from both sides of the equation to get 'x' by itself.
4. Solved for x: We found that x = 5.
5. Verified the Solution: We plugged x = 5 back into the original equation and confirmed that it is correct.

Why This Matters

Solving equations like this is a fundamental skill in algebra. It’s not just about getting the right answer; it’s about understanding the process. These skills come in handy in many areas, not just in math class. Whether you’re figuring out a budget, planning a project, or even coding, knowing how to solve equations can be super useful. Plus, it's like a mental workout that keeps your brain sharp!

Tips for Solving Equations

Here are a few tips to keep in mind when you’re tackling equations:

• Stay Organized: Keep your work neat and organized. Write down each step clearly so you can easily follow your logic and spot any mistakes.
• Combine Like Terms: Always start by simplifying the equation as much as possible by combining like terms.
• Do the Same to Both Sides: Remember, whatever operation you perform on one side of the equation, you must do the same on the other side to maintain balance.
• Check Your Work: It’s always a good idea to plug your solution back into the original equation to make sure it’s correct.
• Practice Makes Perfect: The more you practice, the better you’ll become at solving equations. So, keep at it!

Real-World Applications

You might be wondering, “When am I ever going to use this in real life?” Well, you’d be surprised! Here are a few examples:

• Cooking: If you’re doubling a recipe, you need to adjust the quantities of the ingredients. This often involves solving simple equations.
• Finance: Calculating interest, figuring out loan payments, or balancing a budget all involve equations.
• Science: Many scientific formulas are equations that you need to solve to find unknown quantities.
• Engineering: Engineers use equations all the time to design structures, machines, and systems.

Common Mistakes to Avoid

• Forgetting to Distribute: If you have a term multiplied by a parenthesis, make sure to distribute it to all terms inside the parenthesis.
• Combining Unlike Terms: Only combine terms that are alike (e.g., 'x' terms with 'x' terms, constants with constants).
• Incorrectly Applying Operations: Double-check that you’re performing the correct operations on both sides of the equation.
• Dropping Negatives: Be careful with negative signs. They can easily trip you up if you’re not paying attention.
• Not Checking Your Work: Always verify your solution by plugging it back into the original equation.

Practice Problems

Want to put your skills to the test? Try solving these equations:

1. $$2x + 5 = 11$$

2. $$\frac{1}{2}x - 3 = 4$$

3. $$3(x + 2) = 15$$

Work through them step by step, and remember to check your answers! Solving these kinds of problems is like training your brain muscles—the more you do it, the stronger they get. So, don't be afraid to make mistakes; that's how we learn and improve.

Conclusion

So, there you have it! We’ve successfully solved the equation $\frac{1}{4}x + 4 + \frac{3}{4}x - 2 = 7$ and found that x = 5. Remember, the key to solving equations is to stay organized, combine like terms, isolate the variable, and always check your work. Keep practicing, and you’ll become a pro at solving equations in no time! Keep up the awesome work, guys, and happy solving!