Problem 58 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. Sixteen less than some number is forty-two.

Step-by-Step Solution

Answer

Question: Translate the phrase "Sixteen less than some number is forty-two" into a mathematical equation. Answer: (x - 16) = 42

Step 1: Identify the unknown number and assign a variable

Let "x" represent the unknown number.

Step 2: Interpret the phrase "sixteen less than"

The phrase "sixteen less than" means that we need to subtract 16 from the unknown number. So, the mathematical expression will be (x - 16).

Step 3: Translate the entire phrase into a mathematical equation

The given phrase is "Sixteen less than some number is forty-two". Using the expressions we've derived so far, we can now represent this as an equation: (x - 16) = 42

Key Concepts

Understanding Unknown Variables

In mathematics, unknown variables are symbols that stand in for values that we don't yet know. Think of them as placeholders in a mathematical statement. We usually use letters like \( x \), \( y \), or \( z \) to represent these unknown values. In a problem where you're asked to find an unknown number, choosing a variable is your starting point.

Here's why unknown variables are handy:

• They make it easy to formulate and solve problems.
• They allow you to build equations that can be used to find these unknown values later.
• They provide a clear representation of a mathematical relationship.

In our example, let "\( x \)" denote some unknown number. First, identify that \( x \) is something we need to find, and from the phrase "Sixteen less than some number is forty-two," it is clear that \( x \) is the number we are trying to determine.

Breakdown of Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (like addition, subtraction, multiplication, and division) that define a certain operation or calculation. When translating a phrase into a mathematical expression, it helps to break down each part of that phrase.

Consider the expression derived from "sixteen less than some number":

• "Some number" is initially represented as \( x \), our unknown variable.
• "Sixteen less than" indicates that you should subtract 16 from this unknown number, leading to the expression \( x - 16 \).

This expression \( x - 16 \) clearly indicates an operation where 16 is subtracted from the unknown number \( x \). It’s a simple yet powerful way to concisely represent the entire phrase in math terms.

Crafting Equations from Mathematical Phrases

Equations are more than just mathematical expressions—they show a relationship between two expressions, separated by an equals sign "=". They are statements that assert the equality of two expressions.

To translate our phrase into an equation, we consider both the left and right sides of the equation:

• The left side, derived from "sixteen less than some number," is represented by our previously derived expression, \( x - 16 \).
• The right side of the equation, "is forty-two," simply translates to the number 42.

By putting everything together, the equation is formed as \( (x - 16) = 42 \). This equation provides a complete depiction of the relationship described in the original phrase. Solving this equation will indeed find the value of the unknown variable \( x \), which was our primary goal: determining the number described in the problem statement.