Problem 46 - College Algebra

Question

evaluate each algebraic expression for the given value of the variable or variables. $$ \frac{7(x-3)}{2 x-16} ; x=9 $$

Step-by-Step Solution

Answer

The evaluated algebraic expression becomes 21.

Step 1: Substitute the Variable

Replace every instance of 'x' in the expression with the given value. Here we substitute \(x = 9\). The expression now becomes \(\frac{7(9 - 3)}{2 * 9 - 16}\).

Step 2: Simplify the Expression

Perform the operations in the correct order, following the BODMAS/BIDMAS rule - brackets, orders (powers and roots), division and multiplication, and addition and subtraction. Simplifying within the brackets, we get \(\frac{7 * 6}{2 * 9 - 16} = \frac{42}{18 - 16} = \frac{42}{2}\).

Step 3: Final Result

Complete the division to get the final evaluated expression. So, \(\frac{42}{2} = 21\).

Key Concepts

Substitute Variables in Algebra

To evaluate algebraic expressions, one of the first steps is to substitute any given values for the variables present in the expression. This means replacing the letter (often x, y, or another alphabet) with a numerical value that is provided. When you do this accurately, you're already on your way to finding the solution.

For example, in the expression \( \frac{7(x-3)}{2x-16} \), if you're given that x equals 9, substitute 9 for every instance of x. This yields a new expression: \( \frac{7(9-3)}{2 \cdot 9 - 16} \). It's crucial to substitute carefully to avoid any errors in the next steps of simplification.

Simplify Algebraic Expressions

Once variables are substituted, the next step is to simplify the expression—essentially streamlining it until we have a clear, concise numerical value or a less complex algebraic expression. This involves carrying out arithmetic operations such as addition, subtraction, multiplication, and division, as well as simplifying any brackets.

Following the substitution step with our example \( \frac{7(9-3)}{2 \cdot 9 - 16} \) leads to the simplification of the bracket first, resulting in \( \frac{7 \cdot 6}{18 - 16} \), and further simplification gives us \( \frac{42}{2} \) because we have followed the operational hierarchy and done the multiplication before the subtraction.

Order of Operations in Algebra

The order of operations in algebra, often remembered by the acronym PEMDAS (parentheses, exponents, multiplication and division, addition and subtraction), guides us in simplifying expressions correctly. It's crucial to follow this sequence to arrive at the correct answer.

In our example, we first assess the expression inside the parentheses: \( 9-3 = 6 \). Then we handle the multiplication: \( 7 \cdot 6 \). Division follows with \( 2 \cdot 9 - 16 \) and finally we end up with the division \( 42 \div 2 \) to land at the correct answer of 21. Not adhering to PEMDAS could lead to a completely different, and incorrect, result!