Problem 31 - Fundamentals of Mathematics

Question

If three less than six times a number is equal to five times the number minus three, what is the number?

Step-by-Step Solution

Answer

The number is 0.

Step 1: Assign a variable

Let's assign a variable to the unknown number. Let \( x \) represent the number we need to find.

Step 2: Set up the equation

Based on the problem, we translate the words into an algebraic equation. The phrase 'three less than six times a number' translates to \( 6x - 3 \) and 'five times the number minus three' is \( 5x - 3 \). Thus, the equation is:\[ 6x - 3 = 5x - 3 \]

Step 3: Simplify the equation

We simplify the equation by first eliminating like terms on both sides. Subtract \( 5x \) from both sides:\[ 6x - 5x - 3 = 5x - 5x - 3 \]This simplifies to:\[ x - 3 = -3 \]

Step 4: Solve for the variable

Add 3 to both sides to solve for \( x \):\[ x - 3 + 3 = -3 + 3 \]This simplifies to:\[ x = 0 \]

Step 5: Verify the solution

Substitute \( x = 0 \) back into the original sections of the problem to ensure both sides of the initial equation are equal:- The left side: \( 6(0) - 3 = -3 \)- The right side: \( 5(0) - 3 = -3 \)Both sides are equal, confirming our solution is correct.

Key Concepts

Understanding Equation Solving

Equation solving in algebra is like a puzzle where we fill in unknowns with numbers that make both sides of the equation equal. The primary goal is to find the value of the unknown variable that satisfies the equation. In our example, the equation given is:

• \[ 6x - 3 = 5x - 3 \]

To solve an equation, you generally need to isolate the variable on one side of the equation. This involves simplifying expressions, collecting like terms, and performing arithmetic operations to keep the equation balanced. Here, subtracting \(5x\) from both sides helped us simplify the equation and move closer to a solution. Displaying these transformations step by step provides clarity and ensures no mistakes are made.

The Importance of Variable Assignment

Assigning a variable is a crucial first step in algebra. It provides a placeholder for the unknown value you're trying to find. In many word problems, you need to translate few pieces of information into algebraic expressions or equations. It's common to use letters like \( x \), \( y \), or \( z \) as variables, but any letter or symbol can be used.
In our problem, we chose to assign the unknown number to \( x \). This choice allows us to set up an equation based on the descriptions in the problem. By translating phrases like 'six times a number' into algebraic expressions like \( 6x \), we make it easier to manipulate and solve the equation. This assignment process is fundamental in solving not just simple equations, but also more complex ones involving multiple unknowns.

The Process of Problem Verification

Once you have found a solution, verifying it ensures that your answer is correct. Verification involves substituting the value back into the original equation to check for equality.
For example, after solving the equation and finding \( x = 0 \), it's essential to plug this solution back into both expressions we derived from the problem:

• Left side: \( 6(0) - 3 = -3 \)
• Right side: \( 5(0) - 3 = -3 \)

Both sides equate to \(-3\), confirming our solution is correct. This step ensures there's integrity in your work and helps catch any errors that might have arisen during calculations. Problem verification is a good habit that reinforces confidence in your mathematical procedures and results.