Solving For X In The Equation A - 2[b - 3(c - X)] = 6 A Comprehensive Guide

Hey guys! Today, we're diving into a fun little algebraic equation where we need to isolate and solve for the variable x. The equation we're tackling is:

a - 2[b - 3(c - x)] = 6

This might look a bit intimidating at first glance with all the parentheses and brackets, but don't worry! We'll break it down step by step, making it super easy to understand. Think of it like peeling an onion – we'll get to the core (which is x) one layer at a time. So, grab your pencils, and let's get started!

Breaking Down the Equation: A Step-by-Step Guide

Step 1: Distribute the Innermost Parentheses

Our first mission is to simplify the expression inside the innermost parentheses. We have -3(c - x). Remember the distributive property? We need to multiply -3 by both c and -x. This gives us:

-3 * c = -3c
-3 * -x = +3x

So, -3(c - x) becomes -3c + 3x. Now, let's rewrite the entire equation with this simplification:

a - 2[b - 3c + 3x] = 6

See? We've already made progress. The equation looks a little less scary now.

Step 2: Distribute the Brackets

Next up, we need to deal with the brackets. We have -2[b - 3c + 3x]. Again, we use the distributive property, multiplying -2 by each term inside the brackets:

-2 * b = -2b
-2 * -3c = +6c
-2 * 3x = -6x

So, -2[b - 3c + 3x] becomes -2b + 6c - 6x. Let's plug this back into our equation:

a - 2b + 6c - 6x = 6

Awesome! We're getting closer. We've removed the brackets and now have a more manageable equation.

Step 3: Isolate the Term with 'x'

Our goal is to get x all by itself on one side of the equation. To do this, we need to isolate the term that contains x, which is -6x. Let's move all the other terms to the right side of the equation. We'll do this by adding or subtracting the terms from both sides.

First, let's get rid of a. We subtract a from both sides:

a - 2b + 6c - 6x - a = 6 - a

This simplifies to:

-2b + 6c - 6x = 6 - a

Next, let's get rid of -2b. We add 2b to both sides:

-2b + 6c - 6x + 2b = 6 - a + 2b

This simplifies to:

6c - 6x = 6 - a + 2b

Finally, let's get rid of 6c. We subtract 6c from both sides:

6c - 6x - 6c = 6 - a + 2b - 6c

This simplifies to:

-6x = 6 - a + 2b - 6c

We've successfully isolated the term with x! Now, we just need to get rid of the coefficient (-6) attached to it.

Step 4: Solve for 'x'

We have -6x = 6 - a + 2b - 6c. To solve for x, we need to divide both sides of the equation by -6:

-6x / -6 = (6 - a + 2b - 6c) / -6

This gives us:

x = (6 - a + 2b - 6c) / -6

We can also rewrite this by distributing the division by -6 to each term in the numerator:

x = 6/-6 - a/-6 + 2b/-6 - 6c/-6

Simplifying each term, we get:

x = -1 + a/6 - b/3 + c

So, we've done it! We've successfully solved for x. The solution is:

x = -1 + a/6 - b/3 + c

Or, you can rearrange the terms to write it as:

x = c + a/6 - b/3 - 1

Checking Our Solution

It's always a good idea to check our solution to make sure we didn't make any mistakes along the way. To do this, we'll plug our solution for x back into the original equation:

a - 2[b - 3(c - x)] = 6

Substitute x = -1 + a/6 - b/3 + c into the equation:

a - 2[b - 3(c - (-1 + a/6 - b/3 + c))] = 6

Now, let's simplify step by step:

First, simplify inside the innermost parentheses:

c - (-1 + a/6 - b/3 + c) = c + 1 - a/6 + b/3 - c = 1 - a/6 + b/3

Now substitute this back into the equation:

a - 2[b - 3(1 - a/6 + b/3)] = 6

Next, distribute the -3:

-3(1 - a/6 + b/3) = -3 + a/2 - b

Substitute this back into the equation:

a - 2[b - 3 + a/2 - b] = 6

Simplify inside the brackets:

b - 3 + a/2 - b = a/2 - 3

Substitute this back into the equation:

a - 2[a/2 - 3] = 6

Distribute the -2:

-2[a/2 - 3] = -a + 6

Substitute this back into the equation:

a - a + 6 = 6

Simplify:

6 = 6

Our solution checks out! This confirms that our solution for x is correct.

Common Mistakes to Avoid

When solving equations like this, there are a few common mistakes that students often make. Let's go over them so you can avoid these pitfalls:

1. Forgetting to Distribute: This is a big one! Remember, when you have a number multiplied by a set of parentheses or brackets, you need to multiply that number by every term inside. For example, in -2[b - 3(c - x)], you need to distribute the -2 to both b and the result of -3(c - x). Similarly, when dealing with the inner parentheses, you need to distribute the -3 to both c and -x.

2. Sign Errors: Watch out for those pesky negative signs! A simple sign error can throw off your entire solution. Pay close attention when you're multiplying or dividing by negative numbers. For instance, -3 * -x is +3x, not -3x.

3. Combining Unlike Terms: You can only combine terms that are