Problem 13 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when \(k=3\) $$ \frac{k}{33} $$

Step-by-Step Solution

Answer

The simplified variable expression when \(k=3\) is \(\frac{1}{11}\).

Step 1: Understand the Problem

In this problem, we need to substitute the given value of \(k\) into the variable expression and simplify.

Step 2: Substitute the Value

We know that \(k=3\). So, everywhere we see \(k\) in the expression, we'll substitute the value of 3. This gives us \(\frac{3}{33}\)

Step 3: Simplify the Expression

Next we simplify the expression. The fraction \(\frac{3}{33}\) reduces to \(\frac{1}{11}\), by dividing both the numerator (3) and the denominator (33) by 3, which is the greatest common divisor of 3 and 33.

Key Concepts

Substitution in Algebra

When working with algebraic expressions, substituting variables with numbers is like swapping out puzzle pieces to see the whole picture. Take your expression involving a variable—for instance, \( k \). To evaluate this expression for \( k=3 \) like in our exercise, we follow the strategy of replacing every \( k \) with the number 3. It's essential to perform this substitution accurately to maintain the integrity of the expression. After substitution, your next step is simplification, which brings us to our next key area.
Substitution is a foundational skill in algebra used to solve equations, evaluate expressions, and understand mathematical relationships. Think of it as giving a specific example to abstract scenarios. A variable might represent anything, but once you substitute a number, it becomes something real you can work with. Just remember that precise substitution leads to correct results and avoid altering the structure of the original expression.

Simplifying Fractions

Simplifying fractions is a way to make them easier to understand and work with. It's like cutting a pie into fewer pieces so each one is bigger, but the pie (or the value of the fraction) is still the same size. Simplification often involves finding the greatest common divisor (GCD) of the numerator (top number) and the denominator (bottom number), then dividing both by that number.

• Identify the GCD of the numerator and denominator.
• Divide both numbers by this GCD.
• Check if the new fraction can be reduced further—it should be in its simplest form where no number except 1 can evenly divide both the numerator and the denominator.

For instance, in our example, the fraction \(\frac{3}{33}\) is simplified by finding that 3 is the GCD of both 3 and 33. After dividing the top and bottom by 3, we get \(\frac{1}{11}\), a fraction that cannot be reduced any further. Simplifying fractions makes them neater and often easier when carrying out further arithmetic operations.

Numerical Expression Evaluation

Evaluation of a numerical expression involves performing the operations, such as addition, subtraction, multiplication, or division, to find a numerical value. It's much like following a recipe, step by step, until you get the final dish. In algebraic expressions, this may include substituting variables with numbers and then simplifying.
To evaluate an expression correctly:

• Perform substitution if the expression includes variables.
• Use the order of operations—remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Simplify the expression, which may include reducing fractions as in our textbook exercise.

In our example, after substituting \( k \) for 3, we end up with \(\frac{3}{33}\). We then simplify this numerical expression down to \(\frac{1}{11}\), which is the final evaluation. Understanding how to evaluate expressions is crucial because it applies to various math fields, from simple arithmetic to advanced calculus.