Problem 17 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ 5 a-2 s, \text { if } a=-5 \text { and } s=1 $$

Step-by-Step Solution

Answer

The value of the expression is -27.

Step 1: Substitute the Given Values

Start by substituting the given values for the variables in the expression. Replace \(a\) with \(-5\) and \(s\) with \(1\) in the expression \(5a - 2s\). This gives us: \[5(-5) - 2(1)\]

Step 2: Perform the Multiplications

Next, perform the multiplications in the expression. Calculate \(5 \times (-5)\) and \(-2 \times 1\): \[5(-5) = -25\] \[-2(1) = -2\]

Step 3: Solve the Expression

Finally, solve the expression by adding the results from Step 2: \[-25 - 2 = -27\]

Key Concepts

Substitution

Substitution is the process of replacing variables in an algebraic expression with actual numeric values. This forms the foundation for evaluating expressions. In our given problem, we have an algebraic expression, \(5a - 2s\), where we need to substitute the variables \(a\) and \(s\) with provided values.
To do this:

• Identify the variables and their corresponding values from the problem statement. Here, \(a = -5\) and \(s = 1\).
• Substitute these values into the expression to replace the variables, giving us the new expression, \(5(-5) - 2(1)\).

Substitution helps transform the expression into a form that can be more easily calculated, acting as a crucial step in bridging the equation to its numerical result.

Arithmetic Operations

Arithmetic operations form the basic building blocks of algebra, and include addition, subtraction, multiplication, and division. When evaluating an algebraic expression, correctly carrying out these operations is critical.
For our expression \(5(-5) - 2(1)\), the operations involved are multiplication and subtraction:

• First, conduct the multiplications: \(5 \times (-5)\) and \(-2 \times 1\).
• This results in \(-25\) from \(5 \times (-5)\) and \(-2\) from \(-2 \times 1\).
• Finally, perform the subtraction: \(-25 - 2\), which simplifies to \(-27\).

Understanding the order in which to perform these operations, following the sequence of multiplication followed by subtraction, ensures accuracy in deriving the correct numerical result from the expression.

Evaluation of Expressions

Evaluation of expressions involves the process of calculating an algebraic expression to determine its numeric value after substitution. Once you replace variables with specific numbers and perform the necessary arithmetic operations, you're left with a straightforward calculation.
In our example, after substituting \(a = -5\) and \(s = 1\) into \(5a - 2s\), performing the multiplications results in \(-25\) and \(-2\). The final step is evaluating the expression by solving \(-25 - 2\), which equals \(-27\). This process confirms the result.

• The key here is accuracy in both substitution and arithmetic to ensure the final value is correct.
• In every evaluation, adherence to arithmetic order—handling multiplications and divisions before additions and subtractions—avoids errors.

Evaluation is not just computation; it's a logical progression from the abstract (variables) to the concrete (numbers), providing the solution based on the given conditions.