Easy Algebra: Solve This Linear Equation

Hey math whizzes and everyone else who’s ever stared at an equation and thought, "What even is this?" Today, we're diving headfirst into the awesome world of algebra to tackle a specific problem that's been making the rounds:

Solve the equation: $\frac{3x-11}{4} - \frac{3-5x}{8} = \frac{x+6}{2}$

Don't let those fractions and variables scare you, guys! We're going to break this down step-by-step, making it as clear as mud... wait, no, as clear as crystal! We'll explore the best strategies to solve this bad boy and find the value of 'x' that makes this equation sing. Whether you're a seasoned algebra pro or just dipping your toes in, stick around. By the end of this, you'll be solving equations like a champ.

Understanding the Equation: Let's Get Our Bearings

Alright, before we jump into solving, let's take a good look at the equation we're working with:

$\frac{3x-11}{4} - \frac{3-5x}{8} = \frac{x+6}{2}$

What we've got here is a linear equation with fractions. A linear equation is basically an equation where the highest power of the variable (in this case, 'x') is 1. No x-squared, no x-cubed, just good old 'x'. The tricky part, or the interesting part depending on how you look at it, is the presence of fractions. These fractions have different denominators (4, 8, and 2). Our main mission, should we choose to accept it, is to find the specific value of 'x' that makes the left side of the equation equal to the right side. It's like finding the key to unlock a puzzle!

Why are these different denominators a hurdle? Well, when you're adding or subtracting fractions, you need them to have the same denominator (a common denominator). Trying to add or subtract fractions with different denominators directly is like trying to mix apples and oranges – it just doesn't work smoothly. So, our first big move is going to be getting rid of these denominators or at least making them all the same.

Key Concepts to Remember:

• Linear Equation: An equation where the highest power of the variable is 1. The general form is $ax + b = 0$, where 'a' and 'b' are constants and $a \neq 0$.
• Fractions: Numbers expressed as a ratio of two integers, where the top number is the numerator and the bottom is the denominator. To add or subtract fractions, they must share a common denominator.
• Common Denominator: A number that is a multiple of all the denominators in a given set of fractions. The least common multiple (LCM) is often the most efficient common denominator to use.

Understanding these basics will make the solving process much more straightforward. We're not just blindly manipulating numbers; we're applying fundamental mathematical principles. So, let's gear up and get ready to simplify this beast!

Strategy 1: Conquering Denominators with the Least Common Multiple (LCM)

Okay, team, let's talk strategy. When you're faced with an equation loaded with fractions like ours, the most common and effective way to simplify it is by eliminating the denominators. How do we do that? We use the Least Common Multiple (LCM) of all the denominators. Think of it as finding the smallest number that all your denominators can divide into evenly.

In our equation, the denominators are 4, 8, and 2. Let's find their LCM:

• Multiples of 4: 4, 8, 12, 16, ...
• Multiples of 8: 8, 16, 24, ...
• Multiples of 2: 2, 4, 6, 8, 10, ...

Looking at these lists, the smallest number that appears in all three is 8. So, our LCM is 8. This is going to be our magic number!

Now, here's the cool part: we're going to multiply every single term in the equation by this LCM (which is 8). Why does this work? Because multiplying both sides of an equation by the same non-zero number keeps the equation balanced. It's like giving both sides of a scale an equal weight – they stay level.

Let's apply it:

$8 \times \left( \frac{3x-11}{4} \right) - 8 \times \left( \frac{3-5x}{8} \right) = 8 \times \left( \frac{x+6}{2} \right)$

Now, let's simplify each term by canceling out factors:

• For the first term: $8 \times \frac{3x-11}{4}$. The 8 and 4 cancel, leaving a 2 on top. So, this becomes $2(3x-11)$.
• For the second term: $8 \times \frac{3-5x}{8}$. The 8s cancel out completely, leaving just $-(3-5x)$. Remember that minus sign, it's important!
• For the third term: $8 \times \frac{x+6}{2}$. The 8 and 2 cancel, leaving a 4 on top. So, this becomes $4(x+6)$.

After multiplying by the LCM and simplifying, our equation transforms from this fraction-filled monster into a much friendlier, simpler linear equation:

$2(3x-11) - (3-5x) = 4(x+6)$

See? No more fractions! This step is crucial for making the rest of the solving process manageable. It takes a bit of careful calculation, but the payoff is huge. We've successfully tackled the denominators and paved the way for isolating 'x'. Keep that LCM strategy in your back pocket – it's a lifesaver for many algebra problems!

Strategy 2: Simplifying and Isolating 'x' - The Home Stretch!

Alright, we've cleared the decks of fractions using our LCM. Now, we're left with this beauty:

$2(3x-11) - (3-5x) = 4(x+6)$

Our next move is to simplify both sides of the equation. This involves using the distributive property. Remember that? It's where you multiply the number outside the parentheses by each term inside.

Let's distribute:

• Left side, first part: $2(3x-11) = (2 \times 3x) - (2 \times 11) = 6x - 22$
• Left side, second part: $-(3-5x)$. This means we multiply both terms inside the parentheses by -1. So, it becomes $-1 \times 3 - (-1 \times 5x) = -3 + 5x$. Watch out for those signs!
• Right side: $4(x+6) = (4 \times x) + (4 \times 6) = 4x + 24$

Now, let's substitute these back into our equation:

$(6x - 22) + (-3 + 5x) = 4x + 24$

Combining like terms on the left side (the terms with 'x' and the constant terms):

• Combine the 'x' terms: $6x + 5x = 11x$
• Combine the constant terms: $-22 - 3 = -25$

So, the left side simplifies to $11x - 25$.

The right side is already simplified: $4x + 24$.

Our equation now looks like this:

$11x - 25 = 4x + 24$

We're getting so close, guys! The goal now is to get all the 'x' terms on one side of the equation and all the constant numbers on the other side. This process is called isolating the variable.

Let's move the 'x' terms first. We can subtract $4x$ from both sides to eliminate it from the right side:

$11x - 4x - 25 = 4x - 4x + 24$

$7x - 25 = 24$

Now, let's move the constant terms. We can add 25 to both sides to get the 'x' term alone on the left:

$7x - 25 + 25 = 24 + 25$

$7x = 49$

We're in the final countdown! To get 'x' completely by itself, we need to undo the multiplication by 7. We do this by dividing both sides by 7:

$\frac{7x}{7} = \frac{49}{7}$

$x = 7$

Boom! We've found our solution. The value of 'x' that satisfies the original equation is 7. Wasn't that satisfying? By systematically simplifying and applying algebraic rules, we turned a complex-looking problem into a simple answer.

Verification: Did We Nail It? (Checking Our Answer)

It's always a good idea, especially in algebra, to verify your solution. This means plugging the value you found for 'x' back into the original equation to make sure both sides are equal. If they are, then you know you've got the right answer. It's like getting a "credit" for your work!

Our original equation was:

$\frac{3x-11}{4} - \frac{3-5x}{8} = \frac{x+6}{2}$

And our proposed solution is $x = 7$.

Let's substitute $x=7$ into the left side (LHS):

LHS = $\frac{3(7)-11}{4} - \frac{3-5(7)}{8}$

LHS = $\frac{21-11}{4} - \frac{3-35}{8}$

LHS = $\frac{10}{4} - \frac{-32}{8}$

Now, simplify the fractions:

LHS = $\frac{5}{2} - (-4)$

LHS = $\frac{5}{2} + 4$

To add these, we need a common denominator (which is 2):

LHS = $\frac{5}{2} + \frac{8}{2}$

LHS = $\frac{13}{2}$

Okay, that's the left side. Now let's evaluate the right side (RHS) with $x=7$:

RHS = $\frac{7+6}{2}$

RHS = $\frac{13}{2}$

Compare LHS and RHS:

LHS = $\frac{13}{2}$

RHS = $\frac{13}{2}$

Since LHS = RHS, our solution $x=7$ is correct! This verification step confirms that we did everything right. It’s super satisfying to see it all line up perfectly.

Conclusion: You've Mastered the Equation!

So there you have it, folks! We took a seemingly intimidating equation with fractions, $\frac{3x-11}{4} - \frac{3-5x}{8} = \frac{x+6}{2}$, and systematically solved it to find that $x = 7$. We used the powerful technique of finding the Least Common Multiple (LCM) to eliminate the denominators, simplifying the equation drastically. Then, we applied the distributive property and combined like terms to isolate the variable 'x'. Finally, we confirmed our answer by plugging $x=7$ back into the original equation, proving that our solution is indeed correct.

This problem showcases fundamental algebraic concepts:

• Clearing Fractions: Using the LCM to create a polynomial equation.
• Distributive Property: Simplifying expressions with parentheses.
• Combining Like Terms: Streamlining expressions by grouping similar terms.
• Isolating the Variable: Using inverse operations to solve for 'x'.
• Verification: Checking your answer for accuracy.

Remember, guys, math is all about breaking down complex problems into smaller, manageable steps. Don't be afraid of fractions or variables; they're just parts of the puzzle. With practice and a clear strategy, you can tackle any algebraic challenge that comes your way. Keep practicing, keep exploring, and you'll become an algebra superstar in no time! The answer we found, $x=7$, corresponds to option (D) in the multiple-choice selection provided. Great job!