Problem 42 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. Three less than some number is equal to twice the number minus six.

Step-by-Step Solution

Answer

The number is 3.

Step 1: Identify Components of the Phrase

Firstly, identify the components of the sentence: 'Three less than some number' and 'twice the number minus six'. Let the unknown number be denoted by a variable, say \( x \).

Step 2: Translate Words to Mathematical Symbols

Translate 'three less than some number' to \( x - 3 \). Translate 'twice the number minus six' to \( 2x - 6 \).

Step 3: Formulate the Equation

The problem states that the two expressions are equal, so set them equal to each other: \( x - 3 = 2x - 6 \).

Step 4: Solve the Equation for the Unknown

To solve the equation, first simplify and isolate \( x \):1. Add 6 to both sides to get \( x + 3 = 2x \).2. Subtract \( x \) from both sides to get \( 3 = x \).

Step 5: Verify the Solution

Substitute \( x = 3 \) back into the original expressions:- 'Three less than some number' becomes \( 3 - 3 = 0 \).- 'Twice the number minus six' becomes \( 2(3) - 6 = 0 \).Both sides are equal, verifying the solution is correct.

Key Concepts

Mathematical translation

When faced with translating sentences to mathematical expressions, we initially identify each mathematical component described in words. The phrase "three less than some number" translates to a mathematical expression by first considering "some number" as a variable, typically represented by a symbol such as \( x \). The phrase "less than" indicates subtraction, so "three less than some number" becomes \( x - 3 \). Another aspect of mathematical translation is recognizing phrases that describe operations on numbers. In the sentence "twice the number minus six," "twice the number" means multiplying the unknown number by two, leading to \( 2x \). The words "minus six" then tell us to subtract 6 from this result, yielding the expression \( 2x - 6 \). It's crucial to be comfortable with altering common phrases into math expressions:

• "Less than" typically means subtraction with the number coming after the subtraction.
• "Twice" or "double" means multiplying by 2.
• "Equal to" becomes the equality symbol (\( = \)).

Equation solving

Once we have the correct mathematical equation, solving it becomes the next step. For the equation \( x - 3 = 2x - 6 \), we need to manipulate it so that we isolate the unknown, \( x \). First, notice that simplifying both sides can help. By adding 6 to both sides, we remove the subtraction on the right side:

• Add 6: \( x - 3 + 6 = 2x - 6 + 6 \) simplifying to \( x + 3 = 2x \).

Next, eliminate terms to solve for \( x \). Subtract \( x \) from both sides to gather all the \( x \) terms together:

• Subtract \( x \): \( x + 3 - x = 2x - x \) leading to \( 3 = x \).

These steps simplify our equation to directly find that \( x = 3 \). The process requires balancing both sides of the equation through simple, logical operations, maintaining balance till the solution is obtained.

Variable manipulation

Variable manipulation is all about isolating the unknowns to find their values. In equations like \( x - 3 = 2x - 6 \), the goal is to have one side of the equation just be the variable with no coefficients. This usually involves:

• Performing additions or subtractions to move terms across the equal sign. For example, adding 6 eliminates the \( -6 \).
• Combining like terms, such as moving all \( x \) terms on one side by subtracting \( x \) from both sides.

Understanding the properties of equality is fundamental here. If you add, subtract, multiply, or divide one thing on one side, you must do the same on the other side of the equation. This maintains the equation's balance. With our example, variable manipulation allowed us to reduce the equation to a simple form: \( 3 = x \). Getting proficient at manipulating terms around equations involves practice and recognizing common techniques, ensuring that each operation taken retains the integrity of the equation.