Problem 63 - Elementary and Intermediate Algebra

Question

Translate each problem to an equation. Do not solve. Seven times what number is \(1596 ?\)

Step-by-Step Solution

Answer

The equation is \(7x = 1596\).

Step 1: Identify the Unknown Number

Let's denote the unknown number by a variable. For convenience, choose the variable to be \(x\).

Step 2: Translate the Problem to an Equation

According to the problem, seven times the unknown number is equal to 1596. This can be translated to the equation: \(7x = 1596\).

Key Concepts

Unknown Variables

Understanding unknown variables is the first step in solving any algebraic problem. An unknown variable is a symbol, often a letter like x, that represents a number we do not know yet. For instance, in the problem, 'Seven times what number is 1596?', we don't know the number. So, we assign it a variable, let’s say x, to represent the unknown number.
Why use variables?

• They make it easier to write and solve problems.
• They help in forming equations that represent real-world situations.

Always remember, it's ideal to choose one letter to represent your unknown variable, making it consistent across all steps in your problem.

Linear Equations

Linear equations are equations of the first degree, which means they involve no variables raised to a power higher than one. Typically, a linear equation looks something like ax + b = c.
In our problem, 'Seven times what number is 1596?', this equation can be represented as 7x = 1596, which is a classic linear equation. Here:

• 7 is the coefficient of x.
• x is the unknown variable.
• 1596 is the constant on the right-hand side.

Linear equations are straightforward to handle and usually involve operations like addition, subtraction, multiplication, or division to isolate the variable and find its value.

Algebraic Translation

Algebraic translation is the process of converting verbal statements into mathematical equations. It involves understanding the language of algebra and knowing how to represent words with algebraic symbols.
Let's break down the original problem statement: 'Seven times what number is 1596?'

• 'Seven times' translates to multiplying by 7
• 'What number' suggests an unknown variable, let's call it x
• 'Is' translates to 'equals to' or =
• '1596' stays as it is a constant number

Putting these together, you get the equation 7x = 1596. Learning how to translate words into algebra helps in tackling diverse problems in math and applying these skills in real-life scenarios such as business, engineering, and science.