Problem 27 - Algebra 1

Question

Write the verbal sentence as an equation or an inequality. Three times the quantity two less than a number \(x\) is ten.

Step-by-Step Solution

Answer

The number \(x\) is \(\frac{16}{3}\).

Step 1: Identify the Unknown

Identify the unknown variable in the sentence. Here, the variable is 'a number', represented by \(x\).

Step 2: Interpret the Math Operations

The sentence refers to 'two less than a number \(x\)'. This operation is represented mathematically as \(x - 2\). Then it says 'three times the...' which means we have to multiply the previous results by 3, and the mathematical representation becomes \(3(x-2)\).

Step 3: Construct and Solve the Equation

The sentence ends with 'is ten', which indicates the other side of the equation. So the final equation is \(3(x - 2) = 10\). Now solve the equation. First, distribute the 3 to get \(3x - 6 = 10\). Then add 6 to both sides of the equation, getting \(3x = 16\). Finally, divide both sides by 3 to solve for x, which gives \(x = \frac{16}{3}\).

Key Concepts

Variable Representation

In mathematics, understanding how to represent variables is crucial when converting verbal statements into equations. In our problem, the mention of 'a number' or 'an unknown value' signifies the need for a variable to symbolize this value. We often use letters such as \( x \), \( y \), or \( z \) for this purpose. In the given exercise, we chose \( x \) to represent the unknown number. This forms the foundation of our equation because it takes the place of the unknown and allows us to perform further operations.

By using \( x \), we can then express more complex ideas mathematically, such as operations performed on or with the unknown value. This is the first step towards solving any algebraic equation.

Mathematical Operations

Mathematical operations are the actions we perform on numbers or variables. Common operations include addition, subtraction, multiplication, and division. In the exercise provided, we are asked to interpret the operations within the verbal sentence: 'three times the quantity two less than a number \( x \) is ten.' Here's how to break it down:

• 'Two less than a number \( x \)' translates to the operation \( x - 2 \).

Next, the phrase 'three times the...' indicates multiplication by 3, applied to the expression \( x - 2 \), resulting in \( 3(x - 2) \).

Interpreting these correctly is important because it sets up the structure of the equation that we will solve. Different parts of a word problem map directly to specific mathematical operations, and capturing them accurately leads to forming correct equations.

Solving Equations

Once we have structured our equation from the given sentence, the next task is to solve it. Solving equations involves isolating the variable to find its value. Let's walk through this process using our equation \( 3(x - 2) = 10 \).

• Start by distributing the 3 into the parentheses, resulting in \( 3x - 6 = 10 \).
• Next, add 6 to both sides of the equation to cancel out the subtraction, simplifying it to \( 3x = 16 \).
• Finally, divide both sides by 3 to isolate \( x \), giving us \( x = \frac{16}{3} \).

This sequence of steps demonstrates the fundamental principles of equation solving: applying inverse operations, maintaining balance by performing equal operations on both sides, and simplifying to isolate the variable. Understanding this process not only helps with solving specific types of problems but also develops a foundation for tackling more complex algebraic equations in the future.