Problem 57 - Elementary Algebra

Question

Translate the phrases or sentences to mathematical expressions or equations. A number, times four plus seven.

Step-by-Step Solution

Answer

Question: Translate the phrase "a number times four plus seven" into a mathematical expression. Answer: x * 4 + 7

Step 1: Identify the number

Let's call this unknown number "x".

Step 2: Identify the operation

The operation mentioned in the phrase is "times", which means multiplication.

Step 3: Identify the numbers

There are two numbers mentioned in the phrase: four and seven.

Step 4: Translate the phrase into a mathematical expression

Now, we need to put the information together in the correct order. The phrase says "a number (x) times four (4) plus seven (7)". So the mathematical expression would be: x * 4 + 7

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that use numbers, variables, and operation symbols to represent a quantity. They do not have equal signs like equations, making them tools to express relationships. In our problem, creating an algebraic expression helps us to compactly write the given scenario. Instead of writing out the whole phrase "a number times four plus seven", we construct the expression \( x \times 4 + 7 \). Here, each component of the phrase is represented in a single, cohesive formula.
Algebraic expressions are ideal for solving problems and understanding mathematical relationships because they simplify and condense complex ideas.

Mathematical Translation

Mathematical translation involves converting words and phrases into mathematical symbols and expressions. It’s like learning a new language, where words like "times" and "plus" correspond to operations such as multiplication and addition.
For our exercise, translating the phrase "a number, times four plus seven" requires identifying mathematical equivalents for each word. "A number" becomes a variable \( x \), "times" translates to the multiplication symbol \( \times \), and "plus" translates to the addition symbol \( + \). This skill is crucial as it allows you to transform real-world problems into solvable mathematical equations.
In practice, mathematical translation aids in breaking down complex scenarios into simpler expressions.

Arithmetic Operations

Arithmetic operations are core concepts in mathematics, including addition, subtraction, multiplication, and division. They form the backbone of mathematical calculations and translations. In our exercise, two key operations are identified: multiplication and addition.
"Times" indicates multiplication. So, when we say "a number times four," we perform the operation \( x \times 4 \). Similarly, "plus" means addition, which we use to add seven after multiplying. Thus, we update our expression to \( x \times 4 + 7 \). This simple use of operations shows how each directive in a problem statement affects the overall expression.
Understanding arithmetic operations is fundamental to accurately translating and solving word problems.

Identifying Variables

Identifying variables is a crucial step in translating word problems. Variables are symbols used to represent unknowns or quantities that can change. In algebra, we often use letters like \( x \), \( y \), or \( z \) to denote them.
In our problem, "a number" signifies an unknown, which we represent by the variable \( x \). Assigning variables helps manage and simplify problems since it provides a clear placeholder for unknown values. These placeholders make it easier to build algebraic expressions or equations to solve the problem.
Successfully identifying what your variable stands for is the first step towards constructing meaningful and solvable mathematical models.