Problem 19 - Algebra 1

Question

Evaluate the expression for the given value of the variable. \(12 x\) when \(x=5\)

Step-by-Step Solution

Answer

The expression \(\frac{x}{7}+16\) when evaluated at \(x=14\) equals 18.

Step 1: Substitute the given value into the expression

Plug the value of \(x=14\) into the expression \(\frac{x}{7}+16\). So, it becomes \(\frac{14}{7}+16\).

Step 2: Simplify the fraction

Calculate the fraction \(\frac{14}{7}\) which equals 2. So, it simplifies to \(2+16\).

Step 3: Perform Addition

Add 2 and 16 together. The answer to \(2+16\) is 18.

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra, particularly useful for evaluating expressions. It involves replacing variables with their numerical values to simplify an expression and find its value. This method is crucial for understanding more complex topics in mathematics.

In the provided exercise, the substitution method is applied by taking the given value of the variable, which is \( x=14 \), and plugging it into the expression. It's like replacing a placeholder (the variable) with its actual value, similar to how one might substitute ingredients in a recipe. Using this method ensures the expression reflects the real conditions stated in the problem.

Simplifying Fractions

Simplifying fractions is a key skill in algebra that transforms fractions into their simplest form. A fraction is simplified when the numerator (top number) and the denominator (bottom number) are reduced to the smallest possible integers that still maintain the same value of the fraction.

For example, in the expression \( \frac{14}{7} \), the number 14 can be divided evenly by 7, resulting in the simplified fraction of \( \frac{2}{1} \), which is simply 2. Simplifying fractions not only makes calculations easier but also helps in understanding the underlying relationships between numbers in an equation or expression.

Performing Addition

Performing addition is one of the basic arithmetic operations and is often used in algebra to combine numbers into a sum. After substituting and simplifying, you are often left with a simpler expression that requires addition.

In the exercise, after substituting the value of \( x \) and simplifying the fraction, we are left with the simple addition problem \( 2+16 \). This is straightforward: by adding the two numbers together, we arrive at the final solution. Mastering addition, especially within algebraic expressions, is essential as it is a stepping-stone to more complicated operations and problem-solving.