Problem 68 - Elementary Algebra

Question

Use algebraic notation to write the statement "a number divided by eight, plus five, is equal to ten."

Step-by-Step Solution

Answer

Question: Write an algebraic equation that represents the statement, "A number divided by eight, plus five, is equal to ten." Answer: (x/8) + 5 = 10

Step 1: Identify the variable

Let x represent the unknown number.

Step 2: Translate the statement into an algebraic equation

According to the statement, "a number (x) divided by eight, plus five, is equal to ten." This can be written as: (x/8) + 5 = 10

Key Concepts

Algebraic Equations

Algebraic equations are like puzzles that mathematicians solve to find the value of an unknown variable. They consist of numbers, variables, and operations which are set equal to something. In the context of our exercise, the equation is a balanced statement, with the left side showing a number divided by eight and increased by five, and the right side representing the number ten. To maintain balance, whatever operations you perform on one side, you must do to the other.

For instance, if we start with the equation \( \frac{x}{8} + 5 = 10 \) and decide to subtract 5 from the left side to isolate the fraction \( \frac{x}{8} \) , we must also subtract 5 from the right side, giving us \( \frac{x}{8} = 5 \) . Then, by multiplying both sides by 8, you can solve for \( x \) and find the value of the unknown number. This process is the very essence of working with algebraic equations—manipulating them in a way that adheres to mathematical rules to find the unknown value.

Variable Representation

In algebra, variables are symbols, usually letters, that represent unknown values. Selecting an appropriate variable is a simple yet crucial part of the problem-solving process. In the provided exercise, the variable \( x \) has been chosen to represent the unknown number.

The choice of variable does not affect the final answer, as it is simply a placeholder. However, using common letters like \( x \) and \( y \) can make it easier for others to understand the problem, since these are conventionally used to denote unknowns. In more complicated equations involving multiple unknowns, different variables may be used to distinguish between the different unknown values. The beauty of using variables lies in their flexibility; they allow us to create general solutions that can be applied to many different numbers, not just a single case.

Translating Words to Equations

One of the most vital skills in algebra is translating a verbal statement into an algebraic equation. This requires understanding how common words translate to mathematical operations:

• 'a number' suggests the presence of a variable, which we represented with \( x \).
• 'divided by' indicates division, hence the \( / \) symbol.
• 'plus' translates to addition, shown by the \( + \) sign.
• 'is equal to' tells us that the expression before it is equal to the expression after it, which is where the \( = \) symbol is derived from.

By knowing these translations, the statement 'a number divided by eight, plus five, is equal to ten' can be converted to \( \frac{x}{8} + 5 = 10 \). Mastering the skill of translating words to equations is a fundamental part of solving algebraic problems because it bridges the gap between real-world scenarios and mathematical analysis.