Problem 4 - Elementary Algebra

Question

Translate the following into algebraic equations. Twice some number is subtracted from 30 and the result is \(50 .\)

Step-by-Step Solution

Answer

The algebraic equation is \(30 - 2x = 50\).

Step 1: Identify the Unknown

Let's identify the unknown quantity that needs translation into an algebraic expression. In this case, "some number" is the unknown, and we will represent it with a variable, let's say \(x\).

Step 2: Express "Twice Some Number"

Twice a number can be represented as \(2x\), which means multiplying the unknown by 2.

Step 3: Formulate the Expression for 'Twice Some Number is Subtracted from 30'

Here we need to subtract \(2x\) from 30. This can be expressed algebraically as \(30 - 2x\).

Step 4: Formulate the Entire Equation

The problem states that this result (\(30 - 2x\)) equals 50. Therefore, we set up the equation: \(30 - 2x = 50\).

Key Concepts

Variable Representation

Variables are essential tools in algebra that help us represent unknown quantities or values. In real-world problems, these variables are often depicted by letters like \(x\), \(y\), or \(z\). Let's consider the statement, "Twice some number." Here, "some number" is unknown, which we can represent using a variable, such as \(x\). This translates our unknown quantity into a mathematical expression.

Choosing a variable carefully can simplify solving algebraic equations, especially in word problems. It provides clarity and helps you keep track of what each variable means.

To represent a variable:

• Identify the unknown or the quantity you need to solve for.
• Assign it a letter, usually a lowercase letter, like \(x\).
• Use this variable consistently throughout your calculations or equations.

Translating Word Problems

Translating word problems into algebraic equations may initially seem challenging, but breaking it down step by step can make it manageable. Let’s illustrate this with the example from our exercise.

First, we identify keywords: "twice some number," "subtracted from 30," and "the result is 50." Each of these phrases corresponds to a specific mathematical operation.

Here's how you can translate these phrases:

• "Twice some number": This means multiplying the unknown number by 2, resulting in \(2x\).
• "Subtracted from 30": This directs us to subtract the expression \(2x\) from 30, written as \(30 - 2x\).
• "The result is 50": This tells us that the operation equals 50, leading to the equation \(30 - 2x = 50\).

Word problems often use everyday language. Focus on understanding the actions being described to convert them into mathematical operations. Look out for clue words like "sum," "difference," "product," and "quotient," which indicate mathematical processes.

Solving Linear Equations

Once you've translated the word problem into an algebraic equation, the next step is to solve for the variable. We already formulated the equation as \(30 - 2x = 50\). Solving this involves isolating the variable \(x\) to find its value.

Let's break it down:

• Add \(2x\) to both sides of the equation to eliminate the \(x\) term on one side:
• \(30 = 50 + 2x\)
• Next, subtract 30 from both sides to isolate \(2x\):
• \(0 = 20 + 2x\) simplifies to \(-20 = 2x\)
• Finally, divide both sides by 2 to solve for \(x\):
• \(x = -10\)

These linear equation steps reinforce understanding while providing a strategy to reach a solution. Remember: perform the same operation on both sides of the equation to maintain equality. Simplifying at each step helps to make calculations easier and minimizes errors.