Math Problem Statement

simplify: 2[9(A-B) + 7(2B-A)] - 2[3(2B+A) - 2(A+3B) - 5(A+B)]

Solution

Let's simplify the given expression step by step:

The expression is:

[ 2[9(A-B) + 7(2B - A)] - 2[3(2B + A) - 2(A + 3B) - 5(A + B)] ]

Step 1: Simplify inside the first bracket

Expand the terms inside the first set of brackets: 9(AB)=9A9B9(A - B) = 9A - 9B 7(2BA)=14B7A7(2B - A) = 14B - 7A Now, combine these: 9A9B+14B7A=(9A7A)+(9B+14B)=2A+5B9A - 9B + 14B - 7A = (9A - 7A) + (-9B + 14B) = 2A + 5B Thus, the first bracket becomes: 2[2A+5B]2[2A + 5B]

Step 2: Simplify inside the second bracket

Expand the terms inside the second set of brackets: 3(2B+A)=6B+3A3(2B + A) = 6B + 3A 2(A+3B)=2A6B-2(A + 3B) = -2A - 6B 5(A+B)=5A5B-5(A + B) = -5A - 5B Now, combine these: (6B+3A)+(2A6B)+(5A5B)=(3A2A5A)+(6B6B5B)=4A5B(6B + 3A) + (-2A - 6B) + (-5A - 5B) = (3A - 2A - 5A) + (6B - 6B - 5B) = -4A - 5B Thus, the second bracket becomes: 2[4A5B]2[-4A - 5B]

Step 3: Apply the factors outside the brackets

Now multiply the factors outside the brackets: 2[2A+5B]=4A+10B2[2A + 5B] = 4A + 10B 2[4A5B]=8A10B2[-4A - 5B] = -8A - 10B

Step 4: Combine the results

Now combine the two results: (4A+10B)(8A10B)=4A+10B+8A+10B=(4A+8A)+(10B+10B)=12A+20B(4A + 10B) - (-8A - 10B) = 4A + 10B + 8A + 10B = (4A + 8A) + (10B + 10B) = 12A + 20B

Thus, the simplified expression is:

12A+20B\boxed{12A + 20B}


Would you like further clarification on any part of this solution?

Here are five related questions to extend your understanding:

  1. How would you simplify a similar expression with three variables?
  2. What is the distributive property, and how does it apply in this case?
  3. Can this result be factored? If so, how?
  4. What is the general approach to simplify expressions involving nested brackets?
  5. How would the expression change if the sign of one of the terms inside the brackets were different?

Tip: Always simplify inside the brackets first, then apply the multiplication outside the brackets step by step.

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Math Problem Analysis

Mathematical Concepts

Algebra
Simplification
Distributive Property

Formulas

Distributive Property: a(b + c) = ab + ac

Theorems

Distributive Property

Suitable Grade Level

Grades 8-10