Problem 10 - Elementary Algebra

Question

Translate the following into algebraic equations. Six less than some number is ten times the sum of that number and \(5 .\)

Step-by-Step Solution

Answer

The equation is \( x - 6 = 10(x + 5) \).

Step 1: Define the Variables

Let the unknown number be represented by the variable \( x \). This is the number we are trying to find.

Step 2: Translate Verbal Expression to Equation

The phrase "six less than some number" translates to \( x - 6 \). Next, the phrase "ten times the sum of that number and 5" translates to \( 10(x + 5) \).

Step 3: Set up the Equation

Based on the translation, set up the equation as \( x - 6 = 10(x + 5) \). This represents the statement "six less than some number is ten times the sum of that number and 5."

Key Concepts

Variable Definition

In algebra, variables act as placeholders for unknown values, allowing us to formulate and solve problems in a flexible way. Here, the problem refers to an unknown number, which we need to define with a variable. We use variables to simplify expressions and equations. In most cases, a letter such as \( x \) or \( y \) is used. For the given problem, we denote this unknown number as \( x \). Imagine the variable as a box where we can put different numbers until we find the one that works. Defining a variable is always the first step in solving algebraic problems. It helps in converting verbal statements into mathematical language. This critical step of defining a variable sets the stage for translating the rest of the problem into an equation.

Equation Translation

Equation translation is the process of turning a word problem into a mathematical equation. This can often be tricky because it involves understanding the language used in the problem. The goal is to capture the relationships and operations implied by the words.Let's dissect the given phrase: 'six less than some number is ten times the sum of that number and 5.'

• 'Six less than some number': This indicates a subtraction from our unknown number, which we've called \( x \). So it translates to \( x - 6 \).
• 'Ten times the sum of that number and 5': This suggests that we first add 5 to our unknown number, \( x \), and then multiply the entire sum by 10. In algebraic terms, this becomes \( 10(x + 5) \).

By carefully analyzing each part of the sentence, we can translate the entire statement into the mathematical equation \( x - 6 = 10(x + 5) \). Translating equations accurately is crucial, as it lays the groundwork for finding the solution.

Problem-Solving Steps

Once the variable and equation are defined, solving the problem becomes a matter of following systematic steps. Here is a simple breakdown of how to approach solving the equation obtained from the problem:

• Set up the equation: From the translation, our equation is \( x - 6 = 10(x + 5) \).
• Simplify and solve the equation: Start by expanding any expressions: \( x - 6 = 10x + 50 \). Then, use algebraic manipulation to isolate \( x \). Move all terms involving \( x \) to one side and constant terms to the other. In this step, you might subtract \( 10x \) from both sides and add \( 6 \) to both sides.
• Check your solution: Once you've found a value for \( x \), plug it back into the original equation to ensure both sides equal. This confirms you've solved the problem correctly.

By following these steps, you can systematically approach and solve algebra problems with confidence. Each step methodically builds on the previous one, ensuring that we thoroughly understand and correctly solve the problem.