Problem 66 - Prealgebra

Question

Evaluate each expression if \(a=5, b=1,\) and \(c=3\) $$5 a-6 c$$

Step-by-Step Solution

Answer

The value of the expression is 7.

Step 1: Substitute Values

First, substitute the values given for the variables in the expression. Replace \(a\) with 5 and \(c\) with 3. The expression becomes: \[5(5) - 6(3)\]

Step 2: Multiply Terms

Next, perform the multiplication for each term. Calculate \(5 \times 5\) which equals 25, and calculate \(6 \times 3\) which equals 18. The expression is now: \[25 - 18\]

Step 3: Simplify the Expression

Finally, subtract the second term from the first: \(25 - 18 = 7\).

Key Concepts

Expression Evaluation

Expression evaluation is the process of simplifying an algebraic expression by performing the operations indicated. This involves substituting the given values of variables into the expression, calculating any required multiplications or divisions, and finally simplifying by adding or subtracting. In the exercise, you evaluate the expression \(5a - 6c\) by initially placing known values into the variables \(a\) and \(c\), which are 5 and 3, respectively. Thus, the expression becomes \(5 \times 5 - 6 \times 3\). This is performed step-by-step to ensure accuracy and understanding.

Substitution Method

The substitution method is a crucial technique in algebra where you replace variables with their given or calculated numeric values. This is especially useful in evaluating expressions, as seen in our exercise.

• Identify variables: Determine what each letter in the expression stands for, such as \(a=5\) and \(c=3\).
• Substitute: Replace each variable with its corresponding number, converting the algebraic expression to a numerical one.

This conversion simplifies the overall process because instead of dealing with abstract symbols, you work directly with numbers, making arithmetic operations easier to handle. This step not only aids in accuracy but also helps build confidence with numeric manipulations.

Arithmetic Operations

Arithmetic operations are fundamental mathematical procedures such as addition, subtraction, multiplication, and division. In the context of expression evaluation, these operations are used to break down and solve complex expressions after variables have been substituted with their values. For instance, in the expression \(5 \times 5 - 6 \times 3\), you use the following steps:

• Multiplication: Calculate the results of \(5 \times 5\) and \(6 \times 3\). These results are 25 and 18, respectively.
• Subtraction: After obtaining the results from multiplication, perform subtraction: \(25 - 18\).

These operations are key to simplifying the expression to yield the final result, which is 7. Each operation follows a logical sequence, ensuring clarity and correctness in solving algebraic expressions.