Simplify the Expression: a+b+c-(a-b-c) Step by Step

Distributing Negatives with Algebraic Expressions

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:03 Always solve parentheses first
00:10 Note, negative times positive is always negative
00:13 Note, negative times negative is always positive
00:25 Collect like terms
00:38 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+c(abc)=? a+b+c-(a-b-c)=\text{?}

2

Step-by-step solution

Firstly we need to look at the expression inside of parentheses.

Remember:

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

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

Therefore:

a+b+ca+b+c= a+b+c-a+b+c=

Now let's combine the a a terms:

aa=0 a-a=0

Then we can combine the b b terms:

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

Finally let's combine the c c terms:

c+c=2c c+c=2c

This leaves us with:

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

3

Final Answer

2b+2c 2b+2c

Key Points to Remember

Essential concepts to master this topic
  • Distributive Property: When distributing a negative, flip all signs inside parentheses
  • Technique: -(a-b-c) becomes -a+b+c, changing each term's sign
  • Check: Combine like terms: a-a=0, b+b=2b, c+c=2c gives 2b+2c ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting to distribute the negative to all terms
    Don't just remove parentheses without distributing the negative sign = keeps wrong signs! This leaves terms like -b and -c unchanged instead of becoming +b and +c. 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 negative sign in front of parentheses change all the signs inside?

+

The negative sign acts like multiplying by -1. When you multiply each term by -1, positive terms become negative and negative terms become positive. Think of it as: (abc)=(1)(abc)=a+b+c -(a-b-c) = (-1)(a-b-c) = -a+b+c

How do I remember which signs to flip?

+

Use this rule: opposite of opposite! If you see a negative sign outside parentheses, every term inside gets the opposite of its current sign. Positive becomes negative, negative becomes positive.

What if there are more terms inside the parentheses?

+

The same rule applies! Distribute the negative to every single term, no matter how many there are. For example: (2x3y+4z5)=2x+3y4z+5 -(2x-3y+4z-5) = -2x+3y-4z+5

Can I combine terms before distributing the negative?

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No! You must distribute the negative first, then combine like terms. Changing the order will give you the wrong answer.

How do I check if I distributed correctly?

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Substitute simple numbers for the variables and calculate both the original expression and your simplified version. If they give the same result, you distributed correctly!

What's the difference between +2b+2c and 2b+2c?

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They're exactly the same! The plus sign in front of 2b is understood even when not written. Both expressions are equal and represent the correct answer.

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