Problem 55 - Prealgebra

Question

Find the value of each expression when \(x=-4\) $$\frac{x-14}{36}$$

Step-by-Step Solution

Answer

The value of the expression when \(x = -4\) is \(\frac{-1}{2}\).

Step 1: Substitute x into the expression

First, substitute the given value of \(x\) into the expression. The expression is \(\frac{x-14}{36}\). Replace \(x\) with \(-4\) to get \(\frac{-4-14}{36}\).

Step 2: Simplify the numerator

Now simplify the numerator. Calculate \(-4 - 14\) which equals \(-18\). This changes the expression to \(\frac{-18}{36}\).

Step 3: Simplify the fraction

Simplify the fraction \(\frac{-18}{36}\). Divide both the numerator and denominator by their greatest common divisor, which is 18. This gives \(\frac{-1}{2}\).

Key Concepts

Substitution Method

When it comes to solving algebraic expressions, the substitution method is a handy tool. This technique involves replacing a variable with a given value to simplify the expression. Let's walk through the process with our exercise.

• First, identify the variable in your expression. Here, the expression is \( \frac{x-14}{36} \), where \( x \) is the variable.
• Next, substitute the given value of the variable into the expression. For our exercise, you replace \( x \) with \(-4\).
• After substitution, the expression becomes \( \frac{-4-14}{36} \). This step allows us to remove all variable components and focus purely on numerical computation.

Substitution is particularly useful in evaluating expressions and equations where a specific value for a variable is provided. It helps in transforming complex problems into simpler arithmetic tasks.

Fractions

Fractions are a way to represent parts of a whole and are used extensively in math to represent anything from simple ratios to complex expressions. Given our exercise, the fraction \( \frac{x-14}{36} \) represents the relationship between the numerator \( x-14 \) and the denominator 36. Here's a bit more to understand about fractions:

• Numerator and Denominator: The number above the fraction line is the numerator (part) and the one below is the denominator (whole). So, in \( \frac{-18}{36} \), \(-18\) is the numerator and \(36\) is the denominator.
• Simplifying Fractions: This involves reducing them to their simplest form such that the numerator and denominator have no common factors other than 1.
• Equality of Fractions: Two fractions are equal if they represent the same portion of a whole, like \( \frac{-18}{36} = \frac{-1}{2} \).

Understanding fractions and how to manipulate them is crucial, not just in mathematics but also in real-world applications. Whether it's cooking or calculating proportions, fractions play an essential role in quantitative reasoning.

Simplifying Expressions

Simplifying expressions is about making them as simple as possible without changing their value. This process often includes combining like terms, reducing fractions, and performing arithmetic operations. In our exercise involving \( \frac{-18}{36} \), simplifying the expression involves reducing the fraction to its simplest form. Here's a breakdown of simplifying expressions:

• Identifying Like Terms: Combine terms that contain the same variables or constants. In our problem, it's purely numerical.
• Reducing Fractions: Divide the numerator and the denominator by their greatest common divisor (GCD). For \( \frac{-18}{36} \), that number is 18, resulting in \( \frac{-1}{2} \).
• Performing Arithmetic Operations: Execute any addition, subtraction, multiplication, or division as required. Calculations like \(-4 - 14 = -18\) simplify the expression further before tackling the fraction.

Simplifying expressions helps turn complex arithmetic into more manageable numbers, easing further computation or interpretation. It’s a key method in mathematics to help us solve equations efficiently.