Problem 8 - Elementary Algebra

Question

Translate the following phrases and sentences into mathematical expressions or equations. Seven times two more than a number is twenty-one.

Step-by-Step Solution

Answer

Question: Translate the given phrase into a mathematical equation: "Seven times the sum of two more than a number is twenty-one." Answer: 7(x + 2) = 21

Step 1: Identify the variables and quantities

Let's call the unknown number x. The problem statement says "two more than a number", which means x + 2. It also says "seven times", which means we need to multiply by 7. Finally, "is twenty-one" refers to the equation being equal to 21.

Step 2: Write the equation

Now that we have identified the variables and quantities, we can write the equation as follows: 7(x + 2) = 21

Key Concepts

Variables and Equations

When we talk about variables in mathematics, we're referring to symbols that represent unknown quantities. They are a fundamental aspect of algebra and are typically denoted by letters such as 'x', 'y', 'a', 'b', and so on. In the context of our exercise, 'x' represents an unknown number that we are trying to find.

In any equation, the goal is typically to solve for the variable by manipulating the equation until the variable is isolated on one side. An equation itself is a mathematical statement that asserts the equality of two expressions. It has an equal sign ('=') and tells us that whatever operations we perform on the equation, we must keep it balanced – what we do to one side must also be done to the other. The equation from the exercise, 7(x + 2) = 21, shows that when we multiply 'seven times' some quantity, described as 'two more than a number', the result is twenty-one. This equation can be solved through various algebraic methods, such as distribution and isolation.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, operators, and sometimes symbols that represents a quantity or a relationship between quantities. Unlike equations, expressions don't usually have an equals sign; they're not statements but rather components that can make up an equation.

Understanding how to construct and interpret these expressions is key to solving algebraic problems. In the original exercise, 7(x + 2) is an expression that indicates the operation of multiplying the quantity (x + 2) by 7. Note, however, that expressions are not limited to just numbers and basic operations like addition or multiplication; they can also include more complex operations and functions, such as exponents, square roots, or sine and cosine functions in trigonometry.

Algebraic Translation

Algebraic translation is about turning words into symbols and numbers. For many students, algebraic translation is like learning a new language. It requires recognizing key phrases and understanding the mathematical operations they indicate. When we look at our exercise, phrases such as 'two more than a number' suggest an addition operation, and 'seven times' indicates multiplication.

Mastering the skill of algebraic translation involves several steps:

• Paying attention to the words used and identifying operation keywords such as 'sum', 'product', 'less than', etc.
• Assigning variables to unknown quantities.
• Writing the operations as mathematical symbols next to the variable.
• Constructing the mathematical expression or equation from these symbols.

The ability to translate verbal phrases to algebraic expressions is essential in mathematics and helps in understanding and formulating real-world problems mathematically.