Understanding Algebra: Simplify the Expression a+b-(a-b)

Distributive Property with Negative Signs

a+b(ab)=? a+b-(a-b)=\text{?}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:03 Note, when multiplying negative by positive, the result is always negative
00:10 Note, when multiplying negative by negative, the result is always positive
00:15 Let's gather terms
00:23 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

a+b(ab)=? a+b-(a-b)=\text{?}

2

Step-by-step solution

We begin by addressing the parenthesis first:

Remember that:

When we multiply a positive number by a negative number the result will be negative.

When we multiply a negative number by a negative number the result will be positive.

Hence we obtain the following calculation:

a+ba+b= a+b-a+b=

Next we join together the a coefficients:

aa=0 a-a=0

We then join together the b coefficients:

b+b=2b b+b=2b

We obtain the following:

0+2b=2b 0+2b=2b

3

Final Answer

2b 2b

Key Points to Remember

Essential concepts to master this topic
  • Distributive Rule: Negative sign distributes to all terms inside parentheses
  • Technique: -(a-b) becomes -a+b, so a+b-a+b = 2b
  • Check: Substitute values like a=3, b=2: 3+2-(3-2) = 5-1 = 4 = 2(2) ✓

Common Mistakes

Avoid these frequent errors
  • Not distributing the negative sign to all terms
    Don't just change -(a-b) to -a-b = wrong signs on terms! This keeps the minus in front of b when it should become plus. Always distribute the negative to every single term inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

\( 100-(5+55)= \)

FAQ

Everything you need to know about this question

Why does the minus sign change the signs inside the parentheses?

+

The negative sign acts like multiplying by -1. When you multiply (a-b) by -1, you get -1×a + (-1)×(-b) = -a + b. The negative times negative gives positive!

How do I remember which signs change?

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Think of it as opposite day! Every sign inside the parentheses becomes its opposite when you have a minus sign in front. Positive becomes negative, negative becomes positive.

What if there are more terms inside the parentheses?

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The same rule applies! For example: (2a+3bc)=2a3b+c -(2a + 3b - c) = -2a - 3b + c . Every single term gets its sign flipped.

Can I work from left to right instead?

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You could, but it's riskier! Working with parentheses first (following order of operations) prevents sign errors and makes the algebra much cleaner.

Why do the 'a' terms cancel out completely?

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Because we have +a +a and a -a , which are additive inverses. They always add up to zero: a+(a)=0 a + (-a) = 0 .

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