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Chapter 1: Problem 121

Combine like terms. $$ 13 x-(-2 x)+45-(-21)-7 x $$

Short Answer

Expert verified
\[ 8x + 66 \]

Step by step solution

01

Distribute the Negative Signs

Rewrite the expression to distribute the negative signs: \[ 13x - (-2x) + 45 - (-21) - 7x \] becomes \[ 13x + 2x + 45 + 21 - 7x \]
02

Combine Like Terms for Variables

Combine the terms with the variable \[ 13x + 2x - 7x \] This simplifies to \[ 8x \]
03

Combine Like Terms for Constants

Combine the constant terms \[ 45 + 21 \] This simplifies to \[ 66 \]
04

Write the Final Expression

Combine the results from Step 2 and Step 3: \[ 8x + 66 \]

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Key Concepts

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

Distribute Negative Signs
To simplify expressions, especially when you have negative signs in parentheses, the first thing to do is to distribute those negative signs. This means you need to remove the parentheses by applying the negative sign to the terms inside them. For example, \[ -(-2x) \] becomes \[ +2x \]. Similarly, \[ -(-21) \] becomes \[ +21 \]. This ensures you are working with simpler terms right from the start. For the given problem: \[ 13x - (-2x) + 45 - (-21) - 7x \] distributing the negative signs results in: \[ 13x + 2x + 45 + 21 - 7x \]. This step helps make the expression easier to handle when we start combining like terms.
Combine Like Terms
Now that we have distributed the negative signs, let's focus on combining like terms. Like terms are terms that have the exact same variable raised to the same power. In the expression \[ 13x + 2x + 45 + 21 - 7x \], the variable terms are \[13x, 2x, \text{and } -7x \]. To combine them, simply add or subtract the coefficients (numbers in front of the variables): \[13x + 2x - 7x \]. This simplifies to \[ 8x \]. Next, look at the constants (numbers without variables): \[ 45 \text{and} 21 \]. Adding these together gives \[ 45 + 21 = 66 \]. Combining like terms correctly will significantly simplify your final expression.
Simplify Expressions
After distributing negative signs and combining like terms, the next step is to write the final simplified expression. We have already simplified \[ 13x + 2x - 7x \] to \[ 8x \], and \[ 45 + 21 \] to \[ 66 \]. Bringing these together, the final simplified expression is \[ 8x + 66 \]. Simplifying expressions step by step makes them easier to work with and understand. Remember: distribute negative signs first, combine like terms next, and finally, write the final, simplified expression.

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