Solving A Basic Algebraic Equation

Dec 13, 2025 by ADMIN 35 views

Hey everyone, and welcome back to our math corner! Today, we're diving into a super common type of problem you'll see in algebra: solving for an unknown variable. We've got a cool equation here, $\frac{\frac{3 x}{2}+3}{5}+1=5$ , and our mission, should we choose to accept it (and we totally should!), is to find the value of 'x' that makes this whole thing true. Think of it like cracking a code or solving a puzzle. This kind of problem is foundational, guys, and once you get the hang of it, you'll be zipping through similar ones in no time. We'll break it down step-by-step, making sure we understand each move we make. So, grab your thinking caps, maybe a notebook and pen, and let's get this solved!

Understanding the Equation

Alright, let's really look at the equation we're tackling: $\frac{\frac{3 x}{2}+3}{5}+1=5$ . It might look a little intimidating with those stacked fractions and all, but trust me, it's more manageable than it appears. Our main goal is to isolate 'x'. This means we want to get 'x' all by itself on one side of the equals sign. To do this, we'll use the inverse operations. Remember those? They're like the opposites of the regular operations. Addition's opposite is subtraction, subtraction's is addition, multiplication's is division, and division's is multiplication. We'll apply these in reverse order of operations (PEMDAS/BODMAS) to peel away the numbers and operations surrounding 'x'.

Here's the breakdown of what's happening in the equation:

Inside the innermost parentheses (or implied grouping): We have $\frac{3x}{2}$ . This means 'x' is being multiplied by 3, and then that result is divided by 2.
Next level: To that result ( $rac{3x}{2}$ ), we are adding 3. So, we have $(\frac{3x}{2}+3)$ .
Division: The entire expression $(\frac{3x}{2}+3)$ is then being divided by 5.
Addition: To that result, we are adding 1.
The Grand Finale: All of this equals 5.

Our job is to undo these steps, starting from the outside and working our way in. We want to be super systematic here. Thinking about the order of operations (PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) is key, but when we're solving an equation, we often reverse that order to undo what's been done. So, we'll tackle addition/subtraction first, then multiplication/division, and finally any exponents or parentheses.

Step-by-Step Solution

Let's roll up our sleeves and solve this thing!

Our equation is: $\frac{\frac{3 x}{2}+3}{5}+1=5$

Step 1: Undo the addition of 1.

The first thing we see outside of the main fractional part is '+1'. To undo adding 1, we subtract 1 from both sides of the equation. This is crucial – whatever you do to one side, you must do to the other to keep the equation balanced.

$\frac{\frac{3 x}{2}+3}{5}+1 - 1 = 5 - 1$

This simplifies to:

$\frac{\frac{3 x}{2}+3}{5} = 4$

See? We're already simplifying things down. That '+1' is gone!

Step 2: Undo the division by 5.

Now, look at what's left. The entire numerator, $(\frac{3x}{2}+3)$ , is being divided by 5. The inverse operation of dividing by 5 is multiplying by 5. So, we multiply both sides of the equation by 5.

$5 \times \frac{\frac{3 x}{2}+3}{5} = 4 \times 5$

On the left side, the 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with:

$\frac{3 x}{2}+3 = 20$

We're getting closer! The fraction bar is gone, and we're down to two terms on the left side.

Step 3: Undo the addition of 3.

Next, we need to deal with the '+3' that's sitting right next to our $\frac{3x}{2}$ term. To undo adding 3, we subtract 3 from both sides.

$\frac{3 x}{2}+3 - 3 = 20 - 3$

This leaves us with:

$\frac{3 x}{2} = 17$

Almost there, guys! We've isolated the term containing 'x'.

Step 4: Undo the multiplication by 3 and the division by 2.

Here, 'x' is being multiplied by 3 and then divided by 2. We can undo these in a couple of ways. One way is to multiply by the reciprocal of $\frac{3}{2}$ , which is $\frac{2}{3}$ . Alternatively, we can undo the division first by multiplying by 2, and then undo the multiplication by 3 by dividing by 3.

Let's go with multiplying by 2 first:

$2 \times \frac{3 x}{2} = 17 \times 2$

This gives us:

$3x = 34$

Now, to get 'x' completely by itself, we undo the multiplication by 3 by dividing both sides by 3.

$\frac{3x}{3} = \frac{34}{3}$

And voilà!

$x = \frac{34}{3}$

So, the solution to our equation is $x = \frac{34}{3}$ .

Verification: Checking Our Answer

It's always a super good idea to check your answer, especially in math. This helps you catch any silly mistakes and builds confidence. To verify, we'll plug our solution, $x = \frac{34}{3}$ , back into the original equation and see if both sides are equal.

Original equation: $\frac{\frac{3 x}{2}+3}{5}+1=5$

Substitute $x = \frac{34}{3}$ :

$\frac{\frac{3 \left(\frac{34}{3}\right)}{2}+3}{5}+1$

Let's simplify the numerator first:

$3 \times \frac{34}{3}$ : The 3s cancel out, leaving 34.
So, the numerator becomes $\frac{34}{2}+3$ .
$\frac{34}{2}$ simplifies to 17.
Now we have $17+3 = 20$ .

So, the equation now looks like:

$\frac{20}{5}+1$

Let's simplify further:

$\frac{20}{5}$ is 4.
So, we have $4+1$ .
$4+1 = 5$ .

And there you have it! The left side equals 5, which is exactly what the right side of the original equation is. This means our solution, $x = \frac{34}{3}$ , is absolutely correct. High five!

Why This Matters

Guys, understanding how to solve equations like this is not just about passing a test; it's about building critical thinking skills. These algebraic techniques are the building blocks for more complex math and science concepts. Whether you're figuring out distances, calculating finances, or programming a computer, the logic of manipulating equations is everywhere.

Key takeaways from this problem:

Isolate the variable: Your main goal is to get 'x' by itself.
Use inverse operations: Apply the opposite operation to undo steps.
Balance the equation: Do the same thing to both sides.
Work systematically: Follow the steps carefully, usually reversing the order of operations.
Check your work: Always verify your solution by plugging it back in.

Keep practicing these types of problems, and soon they'll feel like second nature. If you ever get stuck, just remember to break it down, take it one step at a time, and don't be afraid to use those inverse operations. You've got this!

That's all for today! Hope this made sense and was helpful. Let me know in the comments if you have any questions or other equations you'd like us to tackle. See you next time!