Problem 36 - Prealgebra

Question

Evaluate each expression if \(x=9, y=4,\) and \(z=12\) $$(8 y+5)-2 z$$

Step-by-Step Solution

Answer

The expression evaluates to 13.

Step 1: Substitute Variables

Replace each variable in the expression with the given values. Therefore, substitute 4 for \(y\) and 12 for \(z\). The expression becomes \((8 \times 4 + 5) - 2 \times 12\).

Step 2: Calculate Multiplications

Perform the multiplication operations. First calculate \(8 \times 4 = 32\) and \(2 \times 12 = 24\). The expression now is \((32 + 5) - 24\).

Step 3: Compute the Addition

Add 5 to 32 to simplify the expression to \(37 - 24\).

Step 4: Final Subtraction

Subtract 24 from 37 to find the final result, which is 13.

Key Concepts

Substitution

Substitution is the key first step in expression evaluation. It involves replacing variables with their given numerical values. This process is essential because it allows an expression to be simplified into purely numerical terms, making subsequent calculations manageable.

• Identify the variables in the expression. In our example, these are \(y\) and \(z\).
• Replace each variable with its specified value from the problem statement. Here, \(y=4\) and \(z=12\).
• The expression \((8y+5)-2z\) becomes \((8 \times 4 + 5) - 2 \times 12\).

By applying substitution, you transform abstract expressions into concrete numbers, paving the way for systematic computation.

Multiplication

After substitution, perform multiplication, as it sets the stage for simplifying the expression further. When you encounter multiplication in expressions, adhere to the order of operations (PEMDAS/BODMAS): parentheses first, exponents (powers and roots, etc.), then multiplication and division (left to right), and finally addition and subtraction (left to right).

• Calculate each multiplication independently. In our example, we calculate \(8 \times 4 = 32\) and \(2 \times 12 = 24\).
• Replace the multiplication results back into the expression, resulting in \((32 + 5) - 24\).

Multiplication simplifies sections of the expression, allowing you to focus on reducing the equation in a step-by-step manner.

Addition

After performing multiplication, the next step is addition. Adding numbers together might seem straightforward, but its timing in the sequence of operations is crucial.

• Add the results from previous computations together. For the expression \((32 + 5)\), calculate \(32 + 5 = 37\).
• Addition is completed before moving on to subtraction. Make sure additions within parentheses are done first, if applicable.

By executing the addition at this juncture, you prepare the expression for its final simplification through subtraction.

Subtraction

Once addition is completed, it is time to focus on subtraction. This final step leads to the solution. Observing the order of operations ensures accuracy in getting the right answer.

• Subtract the second term from the first. From \(37 - 24\), subtract \(24\) from \(37\), giving \(13\).
• Verify by re-checking each calculation step to ensure consistency with earlier results.

Subtraction concludes the problem-solving sequence, delivering the final evaluated expression value that represents the answer to the problem.