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Chapter 1: Problem 19

Evaluate each following expression for \(x=2, y=5,\) and \(z=3 .\) \(8-(y-x)\)

Short Answer

Expert verified
The expression evaluates to 5.

Step by step solution

01

Substitute the Values

First, we need to substitute the given values into the expression. We replace each variable with its corresponding value: \(x = 2\), \(y = 5\), and \(z = 3\) (though \(z\) is not needed here). The expression becomes: \(8 - (5 - 2)\).
02

Simplify the Inner Expression

Now, simplify the expression inside the parentheses first: \(5 - 2 = 3\). This results in the new expression: \(8 - 3\).
03

Perform the Subtraction

Finally, perform the subtraction: \(8 - 3\). Calculate the result: \(5\).

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Key Concepts

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

Substitution
Substitution is a fundamental concept in algebra, where values are assigned to variables in an expression. This allows us to solve or simplify expressions by replacing variables with known numbers.
In our exercise, we're given the expression \(8 - (y - x)\), and we have specific values for \(x\) and \(y\): \(x = 2\) and \(y = 5\).
Substituting these values into the expression involves the following steps:
  • Identify the variables: Here, the variables in use are \(x\) and \(y\).
  • Replace each variable with its corresponding value: for \(x\), substitute with 2, and for \(y\), substitute with 5.
  • After substitution, the expression becomes \(8 - (5 - 2)\).
Successful substitution sets the stage for simplifying the expression.
Order of Operations
Understanding the order of operations is key to correctly evaluating expressions. It dictates the sequence in which operations should be performed to ensure accurate results.
The common rule to follow is PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
  • Start with computations in parentheses or brackets first.
  • Perform any operations involving exponents or orders next.
  • Handle multiplication and division from left to right.
  • Finally, address addition and subtraction from left to right.
In our problem, after substitution, we have \(8 - (5 - 2)\). According to the order of operations, we deal with the parentheses first, simplifying \(5 - 2\) to 3. This simplification turns the expression into \(8 - 3\), following which the subtraction can occur.
Simplification
Simplification is the process of reducing an expression to its simplest form. This involves performing all possible operations until no further simplifications are available. In algebra, simplicity often refers to having fewer operations or terms.
In our expression, after substituting and following the order of operations, we have \(8 - 3\). This is already close to its simplest form, and performing the subtraction simplifies it fully.
Here's a breakdown of the simplification process:
  • The original complex expression was \(8 - (y - x)\).
  • After substitution and simplifying within the parentheses, it changed to \(8 - 3\).
  • The final step is straightforward arithmetic subtraction, resulting in \(5\).
Simplification not only makes an expression easier to handle but also leads us to the correct solution efficiently.

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