Question

Write each statement using symbols. The sum of \(x\) and 2 is the product of 3 and 4 .

Short answer

\[ x + 2 = 12 \]

Step-by-step solution

Identify the components

First, identify the key components in the given statement. We have: the sum of x and 2, and the product of 3 and 4.

Translate the sum

Next, translate 'the sum of x and 2' into symbols. This is written as: the sum of x and 2: \[ x + 2 \]

Translate the product

Next, translate 'the product of 3 and 4' into symbols. This is written as: the product of 3 and 4: \[ 3 \times 4 \]

Combine the statements

Now, combine the two expressions. 'The sum of x and 2 is the product of 3 and 4' translates into: \[ x + 2 = 3 \times 4 \]

Simplify the product

Finally, simplify the right-hand side of the equation (multiplying 3 and 4): \[ x + 2 = 12 \]

Key concepts

symbolic representation

In algebra, 'symbolic representation' is a way to express statements or problems using symbols and mathematical notations.
It helps simplify and communicate complex ideas clearly.
For example, in the problem 'The sum of \(x\) and 2 is the product of 3 and 4,' we can represent 'the sum of \(x\) and 2' symbolically as \(x + 2\). Similarly, 'the product of 3 and 4' is symbolized as \(3 \times 4\).
By breaking down sentences into mathematical symbols, we can handle and solve problems more easily.
Understanding symbolic representation is essential because it forms the foundation for writing and solving algebraic equations.

equations

An equation is a mathematical statement that asserts the equality of two expressions.
It's written with an equal sign (=) between two expressions showing that they hold the same value.
In the given exercise, 'The sum of \(x\) and 2 is the product of 3 and 4,' translates into the equation \(x + 2 = 3 \times 4\).
This equation states that the sum of \(x\) and 2 is equal to the product of 3 and 4.
Equations play a crucial role in mathematics because they enable us to find unknown values by performing operations on both sides of the equality.

arithmetic operations

Arithmetic operations are basic mathematical procedures which include addition, subtraction, multiplication, and division.
These operations are the building blocks of algebra and are used to manipulate expressions.
In the given exercise, two main arithmetic operations are performed:

• Sum: represented by the plus sign (+). For instance, \(x + 2\) means adding 2 to \(x\).
• Product: represented by the multiplication sign (\( \times \)). For example, \(3 \times 4\) means multiplying 3 by 4.

By simplifying the product, we get \(3 \times 4 = 12\), thus the final equation becomes \(x + 2 = 12\).
Mastery of these basic operations is crucial for solving more complex algebraic problems later on.