Problem 104 - Beginning and Intermediate Algebra

Question

Write each phrase as a mathematical expression using \(x\) as the variable, and simplify. A number multiplied by \(5,\) subtracted from the sum of 14 and eight times the number

Step-by-Step Solution

Answer

14 + 3x

Step 1: Identify the Variable and Constants

The variable in the problem is identified as \(x\). The constants given are 5, 14, and 8.

Step 2: Translate Words to Mathematical Expression

The phrase 'a number multiplied by 5' translates to \(5x\). The phrase 'the sum of 14 and eight times the number' translates to \(14 + 8x\).

Step 3: Set Up the Expression and Subtract

Combine the expressions into the given phrase structure: 'subtracted from the sum of 14 and eight times the number' translates to \(14 + 8x - 5x\).

Step 4: Simplify the Expression

Combine like terms in the expression \(14 + 8x - 5x\): \[14 + (8x - 5x) \Rightarrow 14 + 3x\].

Key Concepts

identifying variables

Variables are symbols that stand in for unknown values in mathematical expressions and equations. They are often represented by letters such as x, y, or z.
In our exercise, the problem tells us to use x as the variable. This x represents 'a number,' or the unknown value that we are working with.
Identifying variables is crucial because it allows us to set up the equation and solve for unknowns.
Always look for phrases that hint at an unknown quantity or value; words like 'a number,' 'an amount,' 'the cost,' and so on are indicators that you will need to use a variable.

constants in algebra

Constants are values that do not change. Unlike variables, which can take on different values, constants stay the same.
In our exercise, the constants are 5, 14, and 8.
Constants are crucial in algebra because they provide fixed numbers that we can perform operations with.

• For example, when the problem mentions '5' in 'a number multiplied by 5', '5' is the constant.
• Similarly, '14' and '8' are constants in 'the sum of 14 and eight times the number.'

Knowing how to identify and use constants keeps the math problem accurate and solvable.

simplifying expressions

Simplifying an expression means combining like terms to make the expression easier to work with.
Let's start with our translated expression: 14 + 8x - 5x.
In algebra, 'like terms' are terms that have the same variable raised to the same power. Here, 8x and -5x are like terms because they both contain the variable x.
To simplify, you combine like terms:

• First, subtract the coefficients of x: 8x - 5x which equals 3x.
• Next, write the simplified expression: 14 + 3x.

Simplifying expressions makes it easier to solve equations and understand relationships between variables and constants.