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Chapter 1: Problem 52

Evaluate each expression if \(\ell=3, w=8,\) and \(s=2\) $$2 \ell+2 w$$

Short Answer

Expert verified
The value of the expression is 22.

Step by step solution

01

Substitute Known Values

Insert the given values for the variables into the expression. We need to evaluate the expression \( 2\ell + 2w \) where \( \ell = 3 \) and \( w = 8 \). Thus, the expression becomes: \( 2(3) + 2(8) \).
02

Calculate Each Term

Perform the multiplication in each term separately. For the first term, calculate \( 2 \times 3 = 6 \). For the second term, calculate \( 2 \times 8 = 16 \).
03

Combine the Results

Add the results of the individual terms together. Add \( 6 \) and \( 16 \) to get the final result: \( 6 + 16 = 22 \).

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Key Concepts

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

Substitution in Algebraic Expressions
Substitution is an essential concept in algebra that involves replacing variables in an expression with known values. This makes problems easier to handle by reducing them to arithmetic operations. When faced with an expression like \(2\ell + 2w\), and given values for these variables (\(\ell=3\) and \(w=8\)), substitution becomes your first step. Simply replace each variable in the expression with its corresponding given value:
  • For \(\ell\), plug in 3, making it \(2(3)\).
  • For \(w\), plug in 8, making it \(2(8)\).

This results in a new numerical expression \(2(3) + 2(8)\), where you can clearly see all variables have been substituted with numbers. Substitution simplifies complex expressions, letting you focus on calculations without worry about variables.
Multiplication in Algebra
Once you have completed substitution in your expression, the next step is to perform multiplication. In our example, after substituting values, we are left with \(2(3) + 2(8)\). Now, let's tackle multiplication:
  • First term: Multiply 2 by 3. This equals 6.
  • Second term: Multiply 2 by 8. This equals 16.

The process of multiplying involves scaling a number by another. In this setting, multiplying captures the idea of repeated addition (e.g., \(2 \times 3\) adds 2 three times). Each of these multiplications provides us with a partial result from the expression, transforming it from variables to tangible numbers.
Adding Results for Final Evaluation
After computing individual products, the next step is to add them together to derive the final value of the expression. The partial results from our multiplications are 6 and 16. Now, let's add these together:
  • 6 (from \(2 \times 3\))
  • 16 (from \(2 \times 8\))
Add them to get: \(6 + 16 = 22\).
This step finalizes the evaluation of the expression initially given, \(2\ell + 2w\). Addition combines quantities to find the sum 鈥� a fundamental arithmetic operation that is vital in combining the calculated terms to reach a solution. This final step in the process ensures that all parts of the equation have been integrated into one comprehensive answer.

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CHALLENGE A counterexample is used to show that a statement is not necessarily true. Draw a counterexample for the statement Supplementary angles form linear pairs.
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