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Chapter 8: Problem 127

Combine like terms. $$ 4 x+7-9 x+12 $$

Short Answer

Expert verified
-5x + 19

Step by step solution

01

Identify like terms

Look at the expression and group the terms that are similar. Like terms are terms that contain the same variable raised to the same power. In the expression \(4x + 7 - 9x + 12\), the like terms are \(4x\) and \(-9x\), and \(7\) and \(12\).
02

Combine the \(x\) terms

Combine the coefficients of the \(x\) terms: \(4x\) and \(-9x\). This results in \(4x - 9x = -5x\).
03

Combine the constant terms

Add the constant terms \(7\) and \(12\). This results in \(7 + 12 = 19\).
04

Write the simplified expression

Combine the results from the previous steps to write the simplified expression: \(-5x + 19\).

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Key Concepts

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

Identify Like Terms
To simplify algebraic expressions, the first step is to identify 'like terms'. Like terms are terms in an algebraic expression that have the same variable raised to the same power. They can be combined because they share this similarity.
In the original expression, \(4x + 7 - 9x + 12\), we have two types of terms: terms with variable \(x\) and constant terms (numbers without any variable).
For this specific example:
  • Variable terms: \(4x\) and \(-9x\)
  • Constant terms: \(7\) and \(12\)
Identifying like terms helps you properly organize the expression for further simplification. It is crucial to recognize these terms accurately to avoid mistakes in the next steps.
Combine Coefficients
After identifying like terms, the next step is to combine their coefficients. Coefficients are the numbers in front of the variables. This step simplifies the terms with the same variable.
To continue with the expression \(4x + 7 - 9x + 12\), we first focus on the variable terms total: \(4x\) and \(-9x\).
  • Combine the coefficients of these terms: \(4 - 9 = -5\)
Thus, the combined term for the variable part is \(-5x\).
Combining coefficients correctly ensures that the algebraic expression maintains its value, but in a simpler form.
Simplify Expressions
The final step in the process is to simplify the expression by adding the constant terms and rewriting the expression neatly.
For our expression, we had already simplified the variable terms to \(-5x\). Now let's focus on the constant terms. We had identified them as \(7\) and \(12\).
  • Adding the constants: \(7 + 12 = 19\)
Now we combine this result with our previously simplified variable term:
\[ -5x + 19 \]
Our fully simplified expression is \(-5x + 19\).
Simplifying expressions helps in making calculations easier and understanding the algebraic relationships more clearly.

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