Problem 86 - Beginning and Intermediate Algebra

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. Twice a number is 17.

Step-by-Step Solution

Answer

The equation is \(2x = 17\).

Step 1: Identify the unknown

We need to represent 'a number' using a variable. In this case, we will use \(x\) to represent the unknown number.

Step 2: Translate 'twice a number'

The phrase 'twice a number' indicates that we need to multiply the unknown number by 2. This can be written as \(2x\).

Step 3: Translate 'is 17'

The word 'is' in mathematics signifies an equal sign. Therefore, 'is 17' can be represented mathematically as \(= 17\).

Step 4: Combine into an equation

Now, combine the two parts: 'twice a number' which is \(2x\) and 'is 17' which is \(= 17\). Therefore, the equation is \(2x = 17\).

Key Concepts

Variables in Equations

In algebra, variables are used to symbolize unknown values, allowing us to form equations that represent real-world scenarios. In the context of our problem, we are looking at a number we don't yet know, which we represent with the variable \(x\). A variable is simply a placeholder for an unknown quantity. By using a variable, we can form relationships and equations that allow us to solve for this unknown value. Think of \(x\) as a box holding a secret number. Whenever we encounter phrases such as 'a number', we can translate them into a variable like \(x\) to help us solve the problem. This makes solving equations more accessible and systematic.

Writing Equations from Sentences

Writing equations from sentences involves converting words into mathematical expressions. This is a crucial skill in algebra, as it helps translate real-world situations into a form that can be manipulated mathematically.

• Begin by identifying key phrases. In the sentence 'twice a number is 17', 'twice a number' suggests multiplication by two.
• The number, which we don't know yet, is represented as \(x\).
• The term 'is' in mathematical language is equivalent to expressing equality, using the equal sign \(=\).

Converting the sentence into an equation, we get \(2x = 17\). This shows the relationship between the unknown number and a specific value, making it possible to communicate and solve the problem mathematically.

Solving Linear Equations

Once we have an equation like \(2x = 17\), our goal is to solve for the variable \(x\). Solving linear equations involves isolating the variable to one side of the equation, making it the subject. To isolate \(x\) in \(2x = 17\), you would:

• Perform the same mathematical operation on both sides of the equation to maintain equality.
• Divide both sides by 2, since \(2x\) indicates two times \(x\).

This gives you:\[\frac{2x}{2} = \frac{17}{2}\]Simplifying, we find that \(x = \frac{17}{2}\). This result tells us the value of the unknown number. Solving linear equations in this way relies on doing operations that "undo" what was done to \(x\), bringing us closer to identifying the initial unknown.