Problem 34 - Algebra 1

Question

Evaluate the expression. $$ \frac{a b}{3 a-10} \text { when } a=8 \text { and } b=7 $$

Step-by-Step Solution

Answer

The result is 4.

Step 1: Substitute the given values

Replace the values of 'a' and 'b' in the given expression. The equation now reads: \( \frac{8 * 7}{3*8 - 10} \).

Step 2: Perform the Multiplications

Perform multiplication operations in the numerator and the denominator independently. The equation will then look like this: \( \frac{56}{24 - 10} \).

Step 3: Subtract in the Denominator

Perform the subtraction in the denominator. The equation becomes: \( \frac{56}{14} \).

Step 4: Perform the Division

Perform the division operation to obtain the final result.

Key Concepts

Substitution in Algebra

Substitution is a fundamental concept in algebra that involves replacing variables in an expression with their given numerical values. This process allows us to evaluate expressions or solve equations efficiently.

When substituting, it is essential to understand which values correspond to each variable. In our exercise, we're working with the variables 'a' and 'b'.

• The problem specifies that \( a = 8 \) and \( b = 7 \).
• We substitute these values directly into the expression \( \frac{ab}{3a-10} \).
• This yields a new expression, \( \frac{8 \times 7}{3 \times 8 - 10} \).

Before we proceed with any calculations, it's crucial to ensure that our substitutions are accurate. This ensures that the entire mathematical process flows correctly, leading to a reliable result.

Order of Operations

The order of operations is a set of rules that dictate the sequence in which operations should be performed to ensure accurate results. In mathematics, especially when dealing with complex expressions, following the order of operations is crucial.

Here's a simple way to remember the sequence: **PEMDAS** (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

In our problem, after substituting the values, the expression becomes \( \frac{8 \times 7}{3 \times 8 - 10} \). To evaluate this correctly, we:

• First, perform the multiplication operations in both the numerator \( (8 \times 7 = 56) \) and within the parentheses in the denominator \( (3 \times 8 = 24) \).
• Next, handle the subtraction in the denominator \( (24 - 10 = 14) \).
• Finally, perform the division \( \frac{56}{14} \), resulting in 4.

By adhering to the correct order, you avoid common mistakes and ensure that the operations yield the correct answer.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. Simplifying makes fractions easier to understand and use in further calculations or comparisons.

In our exercise, the expression results in \( \frac{56}{14} \) after applying the order of operations.

• The task is to simplify this fraction, if possible.
• Identify the greatest common factor (GCF) of the numerator and the denominator, which, in this case, is 14.
• Divide both the numerator (56) and the denominator (14) by the GCF: \( 56 \div 14 = 4 \) and \( 14 \div 14 = 1 \).
• The simplified form is \( \frac{4}{1} \), which equals 4.

Simplifying helps in identifying equivalent fractions and understanding the magnitude of numbers involved in the problem.