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Chapter 7: Problem 57

Simplify. (Assume all denominators are nonzero.) $$ 11+11+x $$

Short Answer

Expert verified
The simplified expression is \(22 + x\).

Step by step solution

01

Identify Like Terms

To simplify the expression, we first identify like terms. Like terms are terms that have the same variables raised to the same powers. In the expression \(11 + 11 + x\), the numbers \(11\) and \(11\) are constant terms (without the variable \(x\)), while \(x\) is a variable term.
02

Combine Like Terms

Combine the constant terms \(11\) and \(11\) together. This is done by simply adding them: \(11 + 11 = 22\).
03

Rewrite the Expression

Rewrite the expression using the simplified term. So, the expression \(11 + 11 + x\) becomes \(22 + x\) after combining the constants.

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Key Concepts

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

Like Terms
When simplifying expressions in algebra, identifying like terms is essential. Like terms are components of an expression that share the same variable and exponent. This allows them to be combined by simple arithmetic. For example, in the expression \(3x + 5 + 7x\), both \(3x\) and \(7x\) are like terms because they both contain the variable \(x\) raised to the same power. Thus, you can combine them to \(10x\).

In contrast, \(5\) is not a like term with \(3x\) or \(7x\) because it lacks the variable \(x\). Recognizing like terms is an important step in simplifying complex expressions quickly.
  • Look for terms with identical variables and exponents.
  • Remember that constants can be like terms with each other.
  • Combine by addition or subtraction depending on their coefficients.
Constant Terms
Constant terms are the parts of an expression that do not change because they do not involve any variables. In other words, they are simply numbers. In our original expression, \(11 + 11 + x\), the two \(11\) terms are considered constant terms. This means that no matter what value you assign to \(x\), the constant parts stay the same.

When simplifying, it is essential to combine all the constant terms first. You add them together just like standard numbers. Here, \(11 + 11\) simplifies to \(22\). Thus, the simplified expression becomes \(22 + x\).
  • Constant terms have no variable associated with them.
  • They can be combined through simple arithmetic.
  • Remain unchanged regardless of variable values.
Variable Terms
Variable terms are components that include variables, signifying an unknown value that can vary. In the expression \(11 + 11 + x\), \(x\) is the sole variable term. Unlike constant terms, variable terms can change depending on what value the variable represents.

When simplifying, variable terms are often left unchanged until they can be combined with like terms. In some expressions, you may have several variable terms that can be added together鈥攕uch as \(3x\) and \(4x\)鈥攖o yield \(7x\). Remember, the algebraic structure of variable terms is crucial in forming equations and algebraic expressions.
  • Variable terms include one or more variables.
  • They introduce flexibility and variability into the equations.
  • Often, these terms remain unaltered until specific values are substituted.

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