Problem 7 - Fundamentals of Mathematics

Question

Twice a number less eight is equal to one more than three times the number.

Step-by-Step Solution

Answer

The number is \( x = -9 \).

Step 1: Understand the Problem

We are tasked with finding a number such that twice the number minus eight equals one more than three times the number.

Step 2: Set Up the Equation

Let the unknown number be represented by \( x \). According to the problem, twice the number minus eight can be expressed as \( 2x - 8 \), and one more than three times the number can be written as \( 3x + 1 \). Set these expressions equal: \( 2x - 8 = 3x + 1 \).

Step 3: Rearrange the Equation

Rearrange the equation to isolate terms involving \( x \) on one side. Subtract \( 2x \) from both sides of the equation to get \( -8 = x + 1 \).

Step 4: Solve for x

Subtract 1 from both sides to solve for \( x \): \(-8 - 1 = x\), which simplifies to \( x = -9 \).

Step 5: Verify the Solution

Substitute \( x = -9 \) back into the original expressions to ensure they balance. \( 2(-9) - 8 = -18 - 8 = -26 \) and \( 3(-9) + 1 = -27 + 1 = -26 \). Both sides are equal, confirming the solution.

Key Concepts

Solving Equations

Solving equations is an important aspect of algebra. It involves finding the value of a variable that makes an equation true. In our exercise, we are given that twice a number minus eight equals one more than three times the number.
This forms the basis of the equation we need to solve.
The goal is to determine the unknown number by manipulating the equation until the variable is isolated.

• Understanding the problem is the first step. Identify what the equation represents and what needs to be solved.
• Setting up the equation correctly helps in finding the right solutions.
• Proceed step by step to simplify the equation.

Once the equation is solved, we can verify it to ensure the solution is accurate. Correct equation solving ensures consistency in mathematical problem solving.

Equation Setup

In algebra, setting up the equation correctly is crucial. In our problem, we need a clear representation of the relationships given in words.
To do this:

• Identify what the variable represents. We use the variable \( x \) to denote the unknown number.
• Translate the verbal expressions into mathematical expressions. "Twice a number less eight" becomes \( 2x - 8 \). "One more than three times the number" translates to \( 3x + 1 \).
• Set the expressions equal to each other: \( 2x - 8 = 3x + 1 \).

This step ensures a clear path to solving the equation and is a critical foundation for isolating the variable.

Variable Isolation

Variable isolation is the process of rearranging an equation to get the variable by itself on one side. It is a key step in solving equations.
Here's how we isolate the variable in our exercise:

• Subtract \( 2x \) from both sides to get all the \( x \) terms on one side. This gives us \( -8 = x + 1 \).
• To isolate \( x \), subtract 1 from both sides, simplifying to \( x = -9 \).

The aim is to have \( x \) alone on one side of the equation. This leaves us with the solution once all computations are done.

Solution Verification

After solving for the variable, it is essential to check your work through solution verification. This ensures the solution is correct.
In our problem, we verify by substituting \( x = -9 \) back into the original expressions.

• Calculate \( 2(-9) - 8 \) to get \(-26\).
• Calculate \( 3(-9) + 1 \) to get \(-26\).

Both sides should match, which they do in this case, confirming that \( x = -9 \) is the correct solution. Verification is a confirmation step that validates our computation and reasoning.