Problem 2 - Elementary Algebra

Question

Translate the following into algebraic equations. When 12 is subtracted from twice some number the result is \(6 .\)

Step-by-Step Solution

Answer

The equation is \( 2x - 12 = 6 \).

Step 1: Determine the Variable

Let's represent the unknown number with a variable. We will choose \( x \) to be this variable.

Step 2: Translate 'Twice Some Number'

The phrase 'twice some number' suggests doubling the unknown number. In algebraic terms, this can be represented as \( 2x \).

Step 3: Translate '12 is Subtracted from'

The phrase 'when 12 is subtracted from' means we subtract 12 from a previous expression. So, from \( 2x \), we subtract 12, which gives us the expression \( 2x - 12 \).

Step 4: Express the Result

The result of subtracting 12 from twice the number is given as 6. Therefore, we equate the expression to 6: \( 2x - 12 = 6 \).

Step 5: Verification

Check that each part of the sentence is correctly translated into the expression. Twice the number is \( 2x \), subtracting 12 gives \( 2x - 12 \), and the result is indeed 6, leading to the equation \( 2x - 12 = 6 \).

Key Concepts

Algebraic Equations

Algebraic equations form the backbone of algebra, providing a way to express relationships and solve problems involving unknown values. They consist of two expressions set equal to each other, often depicted using variables to represent unknown quantities. In the given exercise, the algebraic equation represents the problem as it states that when 12 is subtracted from twice a number, the result equals 6. This relationship is expressed by the equation :

• \( 2x - 12 = 6 \)

This equation highlights the balance between two mathematical expressions, enabling us to ascertain the unknown number by solving it. Understanding algebraic equations enables students to transition from verbal descriptions of problems into a precise mathematical form.

Variable Representation

Variable representation is a fundamental aspect of algebra, allowing us to use symbols to represent unknown values. Variables, typically denoted by letters like \( x \), \( y \), or \( z \), stand in for numbers that we aim to find. In our exercise, the variable \( x \) is used to represent 'some number' which is unknown.

Choosing a variable is a simple but crucial first step. It serves as a placeholder that we manipulate to find solutions to equations. This abstraction is what makes algebra powerful; it lets us perform complex calculations on an unknown quantity and solve for it systematically. Through variable representation, we convert word problems into mathematical language which can then be solved using algebraic techniques.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. Once we have an equation, the next step is to solve it. This often involves performing operations that simplify the equation until the variable is isolated.

• Consider our example equation: \( 2x - 12 = 6 \).

To solve for \( x \), we perform the following steps:

• Add 12 to both sides to get \( 2x = 18 \).
• Then divide each side by 2, resulting in \( x = 9 \).

These steps reflect the core principle of maintaining equality, meaning what you do to one side of the equation, you must do to the other. By doing so, we arrive at the solution of the equation, finding \( x = 9 \), which represents the number originally in the problem. This methodical approach helps students develop a strong grasp of algebraic reasoning and problem-solving.

Algebraic Translation Steps

Algebraic translation is the process of converting language-based descriptions into algebraic expressions and equations. It often begins with identifying key phrases and understanding their mathematical equivalents. In translating the exercise to an equation:

• 'Twice some number' becomes \( 2x \) by multiplying the variable by 2.
• '12 is subtracted from' is translated by subtracting 12 from \( 2x \).
• 'The result is 6' signifies equating the expression to 6, giving us \( 2x - 12 = 6 \).

Breaking down the steps:- Recognize the operation words, such as 'twice,' 'subtract,' and 'result,' which signal multiplication, subtraction, and equality respectively.- Accurately translate these into mathematical terms as shown above.

This systematic approach allows students to convert word problems to equations effectively. It's a crucial skill for tackling more complex algebraic challenges, enhancing both understanding and analytical abilities.