Problem 45 - Fundamentals of Mathematics

Question

Twice a number is added to 5 . The result is 2 less than three times the number. What is the number?

Step-by-Step Solution

Answer

The number is 7.

Step 1: Define the Variable

Let's define the unknown number as \( x \).

Step 2: Translate the Problem into an Equation

According to the problem, twice a number is added to 5: \( 2x + 5 \). This result is equal to 2 less than three times the number: \( 3x - 2 \). Therefore, we can write the equation as: \( 2x + 5 = 3x - 2 \).

Step 3: Rearrange the Equation

Rearrange the equation by getting all terms involving \( x \) on one side and constant terms on the other. Subtract \( 2x \) from both sides: \( 5 = x - 2 \).

Step 4: Solve for the Variable

Add 2 to both sides of the equation to isolate \( x \): \( x = 7 \).

Step 5: Verify the Solution

To verify, substitute \( x = 7 \) back into both expressions: Twice the number plus 5 is \( 2(7) + 5 = 19 \). Three times the number minus 2 is \( 3(7) - 2 = 19 \). Since both expressions evaluate to 19, the solution is correct.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra that involve expressions with one or more variables raised only to the first power. They appear as a line when graphed on a coordinate plane. Linear equations often take the form of \[ ax + b = c \]where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. In the context of real-world problems, such as the one involving finding an unknown number, linear equations help express relationships between quantities in a straightforward manner. Functions are often written in this way to solve for unknown values and understand how different variables interact in a linear way.

Variable Definition

Defining a variable is the first critical step in solving word problems in algebra. A variable is a symbol, often \( x \), \( y \), or \( z \), used to represent an unknown number within an equation. In our exercise, we define the unknown number as \( x \). It stands for the number that, when doubled and added to 5, equals three times the number minus 2.

• It helps translate words into mathematical language.
• Makes equations easier to work with and solve.

In any problem, carefully define your variables based on the context to avoid confusion and ensure solving steps are correctly followed.

Equation Solving

Solving equations is a process of finding the value of variables that make the equation true. Once a linear equation is set up, like \[ 2x + 5 = 3x - 2 \], the next step is to manipulate it to solve for \( x \). Start by moving all terms involving the variable to one side and constants to the other. This usually involves basic arithmetic operations: addition, subtraction, multiplication, or division. In our example, we subtract \( 2x \) from both sides, resulting in\[ 5 = x - 2 \].Then, we add 2 to both sides to isolate \( x \), giving us \[ x = 7 \].Each step simplifies the equation further until the variable is isolated, revealing the solution.

Verification of Solution

Verifying your solution is a crucial part of solving equations to ensure that the value you found is indeed correct. This involves back-substituting the solution into the original equation to check for consistency. Once we found \( x = 7 \), we substitute it back into the two expressions from the problem:

• Twice the number plus 5: \( 2(7) + 5 = 19 \)
• Three times the number minus 2: \( 3(7) - 2 = 19 \)

Both expressions result in 19, confirming our solution is correct. Verification helps prevent errors and increases confidence in your mathematical results.