Problem 45 - College Algebra

Question

evaluate each algebraic expression for the given value of the variable or variables. $$ \frac{5(x+2)}{2 x-14} ; x=10 $$

Step-by-Step Solution

Answer

The evaluated expression for \(x=10\) is \(10\).

Step 1: Substituting the given value for x

Replace \(x\) in the given expression with \(10\), so the expression becomes \(\frac{5(10+2)}{2 \cdot 10-14}\).

Step 2: Simplify the numerator and denominator

Perform the addition in the numerator and the multiplication and subtraction in the denominator separately. This simplification results in \( \frac{5 \cdot 12}{20-14}\). Then, simplify the multiplication in the numerator and subtraction in the denominator to get \( \frac{60}{6}\).

Step 3: Perform the Division

The final step is to divide the numerator by the denominator, which results in \(10\).

Key Concepts

Variable Substitution

In algebra, often the first step to solving an algebraic expression is variable substitution. It's like choosing the right piece for a puzzle. You take a value that is given, such as a specific number for your variable, and you replace the variable in your expression with this value. In our exercise, you were given that the variable \( x \) equals 10. By substituting \( x \) in the algebraic expression \( \frac{5(x+2)}{2x-14} \) with 10, it changes the expression to \( \frac{5(10+2)}{2 \cdot 10 - 14} \).

• The initial expression uses \( x \) as a placeholder.
• Upon substitution, \( x \) becomes 10.

This step sets the stage for the rest of the problem and ensures that we are working with actual numbers rather than variables.

Expression Simplification

Simplification is about making the expression as straightforward as possible. Here, both the numerator and the denominator need to be simplified separately to avoid errors and confusion.Let's look at the numerator \(5(10+2)\):

• First, calculate the contents of the parenthesis, \(10 + 2 = 12\).
• Then multiply this result by 5, giving \(5 \cdot 12 = 60\).

For the denominator \(2 \cdot 10 - 14\), follow similar steps:

• Calculate \(2 \cdot 10 = 20\).
• Then subtract 14, which results in \(20 - 14 = 6\).

After these steps, the simplified form of the expression is \(\frac{60}{6}\). By breaking down each part methodically, you simplify any expression, preparing it for any subsequent operations.

Division in Expressions

Now that we have simplified our algebraic expression to \( \frac{60}{6} \), the last step is straightforward division. Division in expressions can seem daunting, but it's merely finding how many times the denominator fits into the numerator.

• The expression \( \frac{60}{6} \) asks, "How many times does 6 fit into 60?"
• The answer is 10—because \(60 \div 6 = 10\).

Performing division here wraps up the solution. It's crucial to ensure correct division, as the final numeric answer reflects the entire expression's evaluation at the substituted variable's value. Thus, our final answer is 10.