Problem 52 - Algebra for College Students

Question

For Problems \(51-66\), use an algebraic approach to solve each problem. If 1 is subtracted from seven times a certain number, the result is the same as if 31 is added to three times the number. Find the number.

Step-by-Step Solution

Answer

The number is 8.

Step 1: Define the Variable

Let the unknown number be \( x \). This will help translate the word problem into an algebraic equation.

Step 2: Set Up the Equation

From the problem, if 1 is subtracted from seven times the number, it equals 31 added to three times the number. Mathematically, this is expressed as: \( 7x - 1 = 3x + 31 \).

Step 3: Simplify the Equation

First, eliminate the 3x on the right side by subtracting 3x from both sides of the equation: \( 7x - 3x - 1 = 31 \). This simplifies to \( 4x - 1 = 31 \).

Step 4: Isolate the Variable \( x \)

Add 1 to both sides of the equation to get \( 4x = 32 \).

Step 5: Solve for \( x \)

Divide both sides of the equation by 4 to find \( x \). Hence, \( x = \frac{32}{4} = 8 \).

Key Concepts

Algebraic Problem-Solving

Algebraic problem-solving is a systematic process that involves understanding a problem, translating it into a mathematical equation, and solving for the variable. The first step in any algebraic problem is to identify what is being asked.
In our example, the exercise involves finding an unknown number based on certain given conditions.
To tackle this, we need to use algebraic problem-solving strategies, which typically include:

• Identifying unknown elements and expressing them using variables (symbols such as \( x \)).
• Translating verbal statements into algebraic equations.
• Applying mathematical operations such as addition, subtraction, multiplication, and division to manipulate the equations.

The goal of these strategies is to transform a word problem into a solvable equation. This process helps in systematically breaking down complex problems, ensuring each aspect is handled logically and accurately.

Linear Equations

Linear equations are equations of the first degree, meaning they involve only the addition, subtraction, multiplication, and division of variables raised to the first power. A linear equation is typically in the form of \( ax + b = c \). Here, \( a \) and \( b \) are constants, while \( x \) is the variable.
In our given problem, the equation \( 7x - 1 = 3x + 31 \) is a linear equation.
Solving linear equations involves:

• Ensuring the equation realistically represents the problem scenario.
• Using algebraic methods to simplify the equation if necessary.
• Applying techniques to either side of the equation to isolate and solve for the unknown variable.

Linear equations are foundational in algebra because they form the basis for understanding more complex mathematical concepts. They provide a clear pathway from problem statement to solution.

Variable Isolation

Variable isolation is a crucial step in solving equations where the goal is to solve for the unknown variable. It involves rearranging the equation so that the variable of interest is alone on one side of the equation.
In our exercise, we start with the equation \( 7x - 1 = 3x + 31 \). The steps to isolate \( x \) can be broken down as follows:

• Subtract \( 3x \) from both sides, simplifying the equation to \( 4x - 1 = 31 \).
• Add 1 to both sides to further simplify it to \( 4x = 32 \).
• Finally, divide both sides by 4 to isolate \( x \), resulting in \( x = 8 \).

By focusing on what operations affect the variable, we systematically perform inverse operations to "peel away" the influences until the variable is by itself. This process highlights the precision of algebraic methods, ensuring that we find accurate solutions to equations.