Problem 23 - Elementary Algebra

Question

In the following exercises, solve each number word problem. Three times the sum of a number and nine is \(12 .\) Find the number.

Step-by-Step Solution

Answer

The number is \-5\.

Step 1 Title - Set up the equation

Start by setting up an equation based on the problem statement. Let the unknown number be represented by the variable \(x\). According to the problem, three times the sum of a number and nine equals twelve. This can be written as: \[3(x + 9) = 12.\]

Step 2 Title - Simplify the equation

Distribute the 3 in the equation \[3(x + 9) = 12\], resulting in: \[3x + 27 = 12.\]

Step 3 Title - Isolate the variable

Subtract 27 from both sides of the equation to isolate the term containing the variable: \[3x + 27 - 27 = 12 - 27\] \[3x = -15.\]

Step 4 Title - Solve for the variable

Divide both sides of the equation by 3 to solve for \(x\): \[x = \frac{-15}{3}\] \[x = -5.\]

Key Concepts

Solving Equations

Solving equations is like being a detective. You are trying to find the value of an unknown variable. In the word problem, the unknown number is represented by the variable \(x\).
The first step is to set up the equation based on the information given. For example, the problem states three times the sum of a number and nine equals twelve. This translates to: \[3(x + 9) = 12.\]
The objective of solving an equation is to find the value of \(x\). To do this, you'll perform a series of steps to simplify the equation until you have isolated \(x\) on one side.

Distributive Property

The distributive property is a helpful tool in algebra. It allows you to multiply a single term by each term within a parenthesis. In the example, we had \[3(x + 9) = 12.\]
Using the distributive property, you multiply 3 by each term inside the parenthesis: \[3x + 27.\]
This step simplifies the equation and makes it easier to solve. The equation now looks like: \[3x + 27 = 12.\]
Notice how the distributive property helps by removing the parenthesis, breaking the problem into more manageable parts.

Isolating Variables

After applying the distributive property, the next step is isolating the variable. This means getting the variable \(x\) by itself on one side of the equation.
In our example, we have the equation \[3x + 27 = 12.\]
To isolate \(x\), we need to get rid of the constant term on the same side. We do this by subtracting 27 from both sides: \[3x + 27 - 27 = 12 - 27\]
This simplifies to: \[3x = -15,\] bringing us closer to solving for \(x\).

Basic Arithmetic Operations

The final step in solving the equation involves basic arithmetic operations. Here, it means dividing both sides by 3 to solve for \(x\).
From the previous step, we have: \[3x = -15.\]
To isolate \(x\), divide both sides by 3: \[x = \frac{-15}{3}\]
This simplifies to: \[x = -5.\]
Basic arithmetic operations like addition, subtraction, multiplication, and division are foundational skills. They are used in algebra to manipulate equations and find solutions.