Problem 51 - Prealgebra

Question

Four times a number minus 6 is equal to the sum of 3 times the number and \(2 .\) Define a variable and write an equation to find the number.

Step-by-Step Solution

Answer

The number is 8.

Step 1: Define the Variable

To tackle the problem, we first need to define the variable that will represent the unknown number. Let's use the variable \( x \) to represent this unknown number.

Step 2: Translate the Problem into an Equation

Now that we have defined our variable, we need to translate the given word problem into a mathematical equation. The problem states: "Four times a number minus 6 is equal to the sum of 3 times the number and 2." This translates mathematically to the equation: \[ 4x - 6 = 3x + 2 \]

Step 3: Solve the Equation

To solve \( 4x - 6 = 3x + 2 \), we need to get all the terms involving \( x \) on one side of the equation and the constant terms on the other side. Start by subtracting \( 3x \) from both sides: \[ 4x - 3x - 6 = 3x - 3x + 2 \] This simplifies to: \[ x - 6 = 2 \]

Step 4: Isolate the Variable

To isolate \( x \), add 6 to both sides of the equation: \[ x - 6 + 6 = 2 + 6 \] This simplifies to: \[ x = 8 \]

Step 5: Verify the Solution

Substitute \( x = 8 \) back into the original equation to ensure it satisfies the problem: - "Four times a number minus 6" gives: \( 4(8) - 6 = 32 - 6 = 26 \)- "The sum of 3 times the number and 2" gives: \( 3(8) + 2 = 24 + 2 = 26 \)Since both sides equal 26, the solution is verified.

Key Concepts

Solving Equations

When you're faced with an equation like the one in the exercise, the process of solving it can be simplified into systematic steps. Start by aiming to isolate the variable. This means getting the variable by itself on one side of the equation. For the given equation, the steps involve:

• Subtracting or adding terms on both sides to eliminate constants next to the variable.
• Performing the same operation on both sides of the equation, which keeps the equation balanced.
• Continuing with steps until the variable stands alone.

Ultimately, solving equations can feel like solving a puzzle, where the aim is to reveal the value of the unknown variable.

Variable Definition

Defining a variable is the first crucial step in solving algebraic equations, especially when dealing with word problems. A variable acts as a placeholder for the unknown value you are trying to find. In the exercise, we defined the variable \( x \) to represent "the number." Here’s how you define a variable effectively:

• Select a letter to represent the unknown – common choices are \( x \), \( y \), or \( z \).
• Clearly state what the variable stands for in the context of the problem.
• Use this variable consistently throughout the solution process.

By defining a variable clearly, you create a foundation that sets the stage for solving the equation.

Equation Verification

Once you find a solution to an equation, it’s important to verify it. This step ensures that the solution is correct and satisfies the original problem. Verification involves substituting your solution back into the original equation to see if it makes both sides equal.For the exercise:

• Plug the solution \( x = 8 \) back into the equation \( 4x - 6 = 3x + 2 \).
• Calculate each side separately to handle any operations like multiplication, addition, or subtraction.
• Ensure both sides of the equation equal the same value; in this case, both equaled 26.

Verification is the final check to confirm the validity of your solution, providing confidence in your answer.

Translating Word Problems into Equations

Translating word problems into algebraic equations is a critical skill in algebra. It involves converting the language of the problem into mathematical expressions and operations. Let’s break it down using the original problem:

• "Four times a number minus 6" becomes \( 4x - 6 \).
• "The sum of 3 times the number and 2" translates to \( 3x + 2 \).

The key steps include:

• Identifying phrases that indicate mathematical operations, such as "sum" for addition and "times" for multiplication.
• Translating these phrases into mathematical expressions using the defined variable.
• Setting up an equation by equating the mathematical expressions based on the information given in the problem.

By successfully translating a word problem, you pave the way for finding a solution methodically.