Problem 56 - Prealgebra

Question

Find the value of each expression when \(x=-4\) $$\frac{x-12}{36}$$

Step-by-Step Solution

Answer

The value is \( \frac{-4}{9} \).

Step 1: Substitute the Value

To find the value of the expression \( \frac{x-12}{36} \) for \( x = -4 \), substitute \( -4 \) wherever \( x \) appears in the expression. This gives us \( \frac{-4 - 12}{36} \).

Step 2: Simplify the Numerator

Calculate the numerator by combining the numbers inside, which is \( -4 - 12 = -16 \). Now the expression is \( \frac{-16}{36} \).

Step 3: Simplify the Fraction

To simplify \( \frac{-16}{36} \), find the greatest common divisor (GCD) of 16 and 36, which is 4. Divide both the numerator and the denominator by 4: \( \frac{-16 \div 4}{36 \div 4} = \frac{-4}{9} \).

Step 4: State Final Result

The simplified expression is \( \frac{-4}{9} \). This is the value of the expression when \( x = -4 \).

Key Concepts

Substitution

Substitution is a core concept in algebra that involves replacing a variable in an expression with a specific value. This technique allows us to find the numerical value of an expression when certain parameters are known. In the original exercise, we had the expression \( \frac{x-12}{36} \) and were supposed to find its value when \( x = -4 \). To perform substitution:

• Identify the variable(s) in your expression that need replacing.
• Replace each variable with the given number.
• Perform the necessary arithmetic to simplify the expression with the substituted values.

In this instance, substituting \( x = -4 \) led us to the modified expression \( \frac{-4 - 12}{36} \). This is the crucial first step that starts the simplification process.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form where the numerator and the denominator have no common factors other than 1. This step ensures the fraction is as concise as possible, making it easier to work with or interpret. After substituting \( x = -4 \), our fraction was \( \frac{-16}{36} \). To simplify:

• First, ensure your calculation of the numerator and denominator is correct. For us, \(-4 - 12 = -16\).
• Look for any common factors between the numerator and denominator.
• Divide both by their greatest common factor to simplify the fraction.

Finally, the simplified fraction from the original exercise was \( \frac{-4}{9} \). By simplifying fractions, we make them easier to understand and use in further calculations.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that can evenly divide both the numerator and the denominator of a fraction. Finding the GCD is essential for simplifying fractions. In our given exercise, the fraction after substitution was \( \frac{-16}{36} \). To find the GCD:

• List out the factors of each number: \(16\) has factors \(1, 2, 4, 8, 16\); \(36\) has factors \(1, 2, 3, 4, 6, 9, 12, 18, 36\).
• Identify the largest common factor: here it is \(4\).
• Divide both the numerator and the denominator by this GCD.

This ensures the fraction is in its simplest form, as seen where \( \frac{-16}{36} \) became \( \frac{-4}{9} \). Understanding how to find the GCD helps in reducing expressions to their most basic form.