Problem 52 - Algebra 1

Question

Evaluate the expression for the given value(s) of the variable(s). $$\frac{x-5}{6} \text { when } x=30$$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{25}{6} \) when \( x = 30 \).

Step 1: Substitution

Substitute the given value \(x = 30\) into the expression \( \frac{x-5}{6} \). This gives us \( \frac{30-5}{6} \).

Step 2: Simplify the numerator

Subtract 5 from 30 in the numerator to simplify the expression. This gives us \( \frac{25}{6} \).

Step 3: Simplify the fraction

The fraction \( \frac{25}{6} \) cannot be simplified further. Hence this is the evaluated outcome of the expression.

Key Concepts

Substitution Method

The substitution method is an essential concept in algebra. It involves replacing variables in an expression with their actual values. This method is extremely useful when it comes to evaluating expressions, as it makes complex problems understandable and approachable.

For instance, in evaluating the expression \(\frac{x-5}{6}\) when \(x=30\), we start by substituting 30 for \(x\). This transformation changes the abstract expression into a numerical one that can be calculated easily. Using the substitution method not only simplifies the process but also helps to avoid mistakes that might occur when dealing with variables and operations simultaneously.

Practical Steps to Apply the Substitution Method

• Identify the variable in the expression.
• Determine the value that you need to substitute for the variable.
• Replace the variable with its given value.
• Follow the order of operations to evaluate the new numerical expression.

Simplifying Fractions

Simplifying fractions is a fundamental skill in mathematics that makes it easier to work with and understand numbers. A simplified fraction expresses the same value but in a more manageable form, often making it easier to see relationships between numbers or to perform further calculations.

The process involves finding the greatest common divisor (GCD) of both the numerator and the denominator and then dividing both by that number. If no GCD exists other than 1, the fraction is already in its simplest form. For our example, \(\frac{25}{6}\), there is no common divisor for 25 and 6 other than 1, so we say the fraction is already simplified.

Key Points for Simplifying Fractions

• Look for a common factor of both the numerator and denominator.
• Divide both the top and bottom by the greatest common factor.
• Check if the fraction can be further simplified. If not, you have the simplest form.

Numerical Expression

A numerical expression is a mathematical phrase that can contain numbers, operators (like add, subtract, multiply, divide), and sometimes variables that stand in for numbers. To evaluate a numerical expression, you simply perform the operations in the expression using the correct order of operations, which in most countries is remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

The beauty of a numerical expression is that it distills complex relationships down to a form that can be calculated. Returning to our initial problem, \(\frac{x-5}{6}\) becomes purely numerical once we applied the substitution method: \(\frac{30-5}{6}\). After performing the subtraction, we were left with \(\frac{25}{6}\), a simple numerical expression that represented our final answer.