Problem 99 - Elementary and Intermediate Algebra

Question

Complete each table. See Example 11. $$ \begin{array}{|c|c|} \hline s & {\frac{5 s+36}{s}} \\ \hline 1 & {} \\ \hline 6 & {} \\ \hline-12 & {} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The completed table is: 41 for \(s=1\), 11 for \(s=6\), and 2 for \(s=-12\).

Step 1: Evaluate the expression when s=1

Substitute \( s = 1 \) into the expression \( \frac{5s + 36}{s} \). Calculate: \( \frac{5(1) + 36}{1} = \frac{5 + 36}{1} = 41 \).

Step 2: Evaluate the expression when s=6

Substitute \( s = 6 \) into the expression \( \frac{5s + 36}{s} \). Calculate: \( \frac{5(6) + 36}{6} = \frac{30 + 36}{6} = \frac{66}{6} = 11 \).

Step 3: Evaluate the expression when s=-12

Substitute \( s = -12 \) into the expression \( \frac{5s + 36}{s} \). Calculate: \( \frac{5(-12) + 36}{-12} = \frac{-60 + 36}{-12} = \frac{-24}{-12} = 2 \).

Key Concepts

Evaluating Expressions

When dealing with algebra, one of the core concepts you will encounter is evaluating expressions. This means you will take an algebraic expression and find its value by substituting numbers in place of variables. In math, we often use letters like \( s \), \( x \), or \( y \) to represent these variables.
To evaluate an expression, you need to replace the variable with a given number and perform the calculating operations as indicated. Let's walk through the example expression \( \frac{5s + 36}{s} \), step-by-step:

• Identify the variable in your expression, which in this case is \( s \).
• Replace \( s \) with the given number.
• Carry out the operations: multiplication, addition, and division.

As illustrated, we evaluated this expression for different values of \( s \):

• When \( s = 1 \), it evaluates to 41.
• When \( s = 6 \), it evaluates to 11.
• When \( s = -12 \), it evaluates to 2.

Each substitution gives you a specific number which is your answer for that value of the variable.

Substitution Method

Substitution is a technique used to solve expressions or equations, by replacing the variables with the given numbers. It's like filling in the blanks - you take the number that is supposed to replace the letter, plug it into the expression or equation, and solve.
Here's how the substitution method works:

• First, identify what values you need to substitute into the expression. In the example, those values were 1, 6, and -12 for \( s \).
• Every time you see \( s \) in the expression \( \frac{5s + 36}{s} \), replace it with the given number.
• For each substitution, solve the expression carefully, keeping in mind the order of operations (PEMDAS/BODMAS).

This method helps in simplifying the problem by breaking it down into manageable parts. For example, when substituting \( s = 6 \), we calculated \( \frac{30 + 36}{6} = 11 \). By repeating this process for each value of \( s \), you find the resulting values that satisfy the expression.

Rational Expressions

A rational expression looks a lot like a fraction, with polynomials in the numerator, denominator, or both. An expression like \( \frac{5s + 36}{s} \) is called rational because it includes a polynomial in the numerator and a single variable in the denominator.
Rational expressions can be quite useful but need careful handling, especially with the variable in the denominator. Remember the rules:

• The denominator cannot be zero, since division by zero is undefined.
• To evaluate it, substitute numbers in for the variable, then simplify the expression.

For instance, if \( s \) were zero in our example expression, \( \frac{5s + 36}{s} \) would be undefined. But for values like 1, 6, and -12, we substitute these into the expression and perform the required arithmetic operations to get the answer.
Understanding how to work with and simplify rational expressions is critical, as they frequently appear in algebra and higher-level mathematics.