Problem 46 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. A number plus six, divided by two, is seventy-one.

Step-by-Step Solution

Answer

Question: Translate the sentence "A number plus six, divided by two, is seventy-one" into a mathematical equation. Answer: (x + 6) / 2 = 71

Step 1: Identify the variable, operations, and numbers

In the sentence, "A number plus six, divided by two, is seventy-one," the number is unknown, so we can represent it with a variable 'x'. The operations mentioned are addition and division, and the numbers are six and seventy-one.

Step 2: Write the mathematical expression

Based on the sentence, we can write the expression as follows: First, add six to the unknown number 'x', and then divide the result by two. This can be written as: (x + 6) / 2.

Step 3: Form the equation

The sentence states that this expression is equal to seventy-one, so we can now form the equation: (x + 6) / 2 = 71. The final equation, representing the given sentence, is: (x + 6) / 2 = 71.

Key Concepts

Mathematical Expressions

Mathematical expressions are a combination of numbers, variables, and operation symbols that represent a specific value or quantity. In the context of our example, the phrase 'a number plus six, divided by two' is transformed into a numerical expression. In creating such expressions, understanding the order of operations is crucial.

First, variables such as 'x' stand in for unknown numbers, allowing us to describe a general relationship or rule. Next, we interpret the words 'plus' and 'divided by' as operations: addition and division. In the phrase given, 'plus six' indicates we're adding 6 to our unknown number. After this addition, the entire result is to be 'divided by two', which we notate in the expression as a fraction or with a division symbol.

• The word 'plus' translates to the '+' symbol in algebra.
• The phrase 'divided by' translates to '/' or could be shown as a fraction bar.

Once this translation process is complete, we have a clear and concise mathematical representation: \(\frac{x + 6}{2}\) as the expression derived from the phrase.

Algebraic Equations

Algebraic equations are mathematical statements indicating the equality between two expressions, connected by the equals '=' sign. Building from our initial expression, we use the provided information that this quantity 'is seventy-one' to create an equation reflective of this balance.

An equation is a question we're posing, essentially asking, 'What value of x will make this statement true?' The phrase 'is seventy-one' signals that the expression, when evaluated, should yield the number 71. In writing this as an equation, we maintain the expression on one side and place the constant number on the other side of the '=' symbol.

• The word 'is' is essential as it prompts us to use the '=' symbol, indicating equivalence in algebra.

By translating these phrases into the language of algebra, we've constructed the equation \(\frac{x + 6}{2} = 71\), a solvable puzzle waiting for us to find the unknown variable 'x'.

Variables in Algebra

Variables are the letters or symbols in algebra that represent numbers whose values are not yet known or can change. They are the foundational placeholders that make algebra flexible and useful across various problems. In our given problem, the variable 'x' represents the unknown number we're attempting to find.

Using variables allows us to work with abstract concepts and create general rules that are widely applicable, not just to specific numbers. While any letter or symbol can be used as a variable, it is important to be consistent once one is chosen.

• In our case, 'x' is a common choice for a variable in algebra.
• Variables give us the ability to easily manipulate and solve for unknown quantities.

Understanding how to work with variables and translate phrases into expressions and equations is an essential algebra skill, preparing students to tackle a wide range of mathematical problems.