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Chapter 5: Problem 76

Add or subtract as indicated. $$ (8 a b+2 a-3 b)-(6 a b-2 a+3 b) $$

Short Answer

Expert verified
2ab + 4a - 6b

Step by step solution

01

Write down the expression

Start by clearly writing down the given expression: (8ab + 2a - 3b) - (6ab - 2a + 3b)
02

Distribute the negative sign

Distribute the negative sign across the terms in the second set of parentheses: (8ab + 2a - 3b) - 6ab + 2a - 3b
03

Combine like terms

Combine like terms together by grouping them: (8ab - 6ab) + (2a + 2a) + (-3b - 3b) This results in: 2ab + 4a - 6b

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
In algebra, combining like terms helps simplify expressions. Like terms have the same variables raised to the same powers.
For example, in the expression 8ab + 2a - 3b, '8ab' and '6ab' are like terms because they both contain 'ab'. Similarly, '2a' is like with '-2a', and '-3b' is like with '+3b'.
To eliminate confusion, it's useful to group like terms together before combining.
When you combine like terms, you add or subtract their coefficients while keeping the variable part the same. For instance:
  • 8ab - 6ab = (8-6)ab = 2ab
  • 2a + 2a = (2+2)a = 4a
  • -3b - 3b = (-3-3)b = -6b

    • Mastering this skill makes it easier to handle more complex algebraic expressions.
Distributive Property
The distributive property allows you to multiply a term across terms within parentheses. It's expressed mathematically as: ewline a(b + c) = ab + ac.
In this particular problem, we use the distributive property to distribute a negative sign across an expression. For instance, ewline - (6ab - 2a + 3b) becomes -6ab + 2a - 3b.
Essentially, you multiply each term inside the parentheses by -1. This changes the sign of each term:**
  • - (6ab) = -6ab
  • - (-2a) = +2a
  • - (+3b) = -3b

    • After distributing the negative sign, you can then easily combine like terms.
Subtraction of Expressions
Subtracting algebraic expressions might initially look tricky, but it gets simpler with the right approach. This exercise demonstrates subtraction by dealing with parentheses. Starting with the given expressions, ewline (8ab + 2a - 3b) - (6ab - 2a + 3b), you handle the subtraction by distributing the negative sign first. ewline After distribution, you are left with: ewline (8ab + 2a - 3b) - 6ab + 2a - 3b.
From here, follow the steps for combining like terms:
  • (8ab - 6ab)
  • (2a + 2a)
  • (-3b - 3b)
This simplification yields ewline 2ab + 4a - 6b.
Remember the key steps:
  • Distribute any negative signs
  • Combine like terms.

    • Practice and familiarity make these kinds of problems much more manageable over time.

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