Question

Multiple Choice Which of the following represents"the product of 5 and \(x+3\) equals \(6 " ?\) (a) \(5+(x+3)\) (b) \(5 \cdot x+3=6\) (c) \(5(x+3)=6\) (d) None of these

Short answer

The correct choice is (c).

Step-by-step solution

- Understand the Problem

The problem asks for the representation of 'the product of 5 and (x+3) equals 6'.

- Define Terms

'Product' in mathematics means multiplication. Therefore, the problem is describing the multiplication of 5 and the expression (x+3).

- Translate the Expression

To represent 'the product of 5 and (x+3)', we write it as: 5(x + 3)

- Set the Equation

The problem states that this product is equal to 6. Therefore, we write: 5(x + 3) = 6

- Identify the Correct Option

From the given choices, option (c) is 5(x + 3) = 6, which matches our expression.

Key concepts

Product in mathematics

In mathematics, the term 'product' refers to the result of multiplying two or more numbers or expressions. To illustrate, if we multiply 2 and 3, the product is 6. When you see the word product in problems, it's a signal to use multiplication.
In the given exercise, 'the product of 5 and \(x + 3\)' means we need to multiply 5 by the expression \(x + 3\). This is written as \[5(x + 3) \].
This step is crucial to solving the problem accurately. Understanding that product means multiplication will help you easily set up similar equations in the future.

Translation of expressions

Translating verbal mathematical expressions into algebraic terms is a critical skill in algebra. For instance, the phrase 'the product of 5 and \(x + 3\)' directly translates to the multiplication of 5 and the expression \(x + 3\).
When translating problems:

• Identify keywords like 'sum', 'difference', 'product', and 'quotient'.
• Choose the correct mathematical operation based on these keywords.

Our example used the term 'product', which we translated to multiplication, leading us to write \[5(x + 3) \]. This translation step is essential for setting up the correct equation.

Setting equations

Setting equations involves taking a translated expression and forming an equation that matches the problem's conditions. For our exercise, we translated 'the product of 5 and \(x + 3\)' to \[5(x + 3) \].
The statement 'equals 6' tells us to set this expression equal to 6, forming the equation \[5(x + 3) = 6 \].

Here are steps for setting equations properly:

• Translate the given verbal phrase into an algebraic expression.
• Determine what the expression should equal or represent based on the problem.
• Write the complete equation to reflect these conditions.

By following these steps, you can confidently translate and set equations, just as we did in this exercise.