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Chapter 5: Problem 101

Subtract. $$ (8 a-3 b+c)-(2 a+3 b-4 c) $$

Short Answer

Expert verified
6a - 6b + 5c

Step by step solution

01

Distribute the minus sign

First, distribute the minus sign to each term inside the parentheses: (8a - 3b + c) - (2a + 3b - 4c) = (8a - 3b + c) - 2a - 3b + 4c
02

Group like terms together

Next, group the like terms together to make it easier to combine them: (8a - 2a) + (-3b - 3b) + (c + 4c)
03

Combine the like terms

Finally, combine the like terms: (8a - 2a) = 6a, (-3b - 3b) = -6b, (c + 4c) = 5c Therefore, the expression simplifies to: 6a - 6b + 5c

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Key Concepts

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

Distributing minus sign
When subtracting algebraic expressions, it's important to distribute the minus sign across each term inside the parentheses. Suppose we need to subtract \( (8a - 3b + c) - (2a + 3b - 4c) \). Take the minus sign and distribute it to each term in the second expression.
Note: Distributing a minus sign means changing the sign of each term inside the parentheses.
  • First term: \( 2a \) becomes \( -2a \).
  • Second term: \( 3b \) becomes \( -3b \).
  • Third term: \( -4c \) becomes \( +4c \).
So after distributing the minus sign, the expression now looks like this:
\( (8a - 3b + c) - 2a - 3b + 4c \)
Combining like terms
Once we have distributed the minus sign, the next step is to combine like terms.
First, identify the like terms in the expression. Like terms are terms that have the same variable raised to the same power.
For example:
  • \( 8a \) and \( -2a \) are like terms because they both contain the variable \(a\).
  • \( -3b \) and \( -3b \) are like terms because they both contain the variable \(b\).
  • \( c \) and \( 4c \) are like terms because they both contain the variable \(c\).
Now, group these like terms together to make the expression clearer:
\( (8a - 2a) + (-3b - 3b) + (c + 4c) \). This grouping will make it easier for us to combine and simplify in the next step.
Simplifying algebraic expressions
The final step is to simplify the algebraic expression by combining the like terms we grouped previously.
  • Combine the \(a\)-terms: \( 8a - 2a = 6a \).
  • Combine the \(b\)-terms: \( -3b - 3b = -6b \).
  • Combine the \(c\)-terms: \( c + 4c = 5c \).
So, after combining all like terms, the simplified expression is:
\( 6a - 6b + 5c \).
This simplified form is easier to interpret and can be used in further calculations, if needed.
Remember, simplifying expressions helps in making complex algebraic problems more manageable!

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