Problem 75 - Elementary Algebra

Question

Translate the following sentences into linear equations and then solve. The difference of \(5 x\) and 6 is equal to \(4 .\)

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Translate the Sentence into a Linear Equation

The problem states: "The difference of \(5x\) and 6 is equal to 4." To translate this into a linear equation, we write it mathematically as: \[5x - 6 = 4\]

Step 2: Isolate the Variable Term

To solve for \(x\), we need to get all \(x\) terms on one side of the equation and constants on the other. Start by adding 6 to both sides of the equation to eliminate the -6 on the left side. This gives us: \[5x - 6 + 6 = 4 + 6\] Simplifying this, we have: \[5x = 10\]

Step 3: Solve for the Variable

Now, divide both sides of the equation by 5 to get \(x\) by itself: \[\frac{5x}{5} = \frac{10}{5}\] Simplifying yields: \[x = 2\]

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is the language through which we describe patterns, relationships, and changes. In this context, variables like \( x \) can be used to represent numbers we don't yet know. This allows us to write and solve equations that describe those patterns and relationships.

A key concept in algebra is the equation, a statement that two expressions are equal. When working with linear equations, we're looking at equations that result in straight-line graphs. These are typically in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. By learning to manipulate and solve these equations, we can solve a variety of real-world problems.

Algebra is foundational in higher mathematics and its methodology is crucial to understanding many aspects of mathematics and science.

Step-by-Step Process of Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. Here are the steps generally followed to solve equations:

• Identify the Equation: Recognize what type of equation you're dealing with. In linear equations, variables have an exponent of one.
• Isolate the Variable: Begin by moving all terms containing the variable to one side and the constants to the other. This is typically done by performing operations that simplify the equation. For instance, if our equation is \( 5x - 6 = 4 \), we can add 6 to both sides to start isolating \( x \).
• Solve for the Variable: Once you have the variable terms alone on one side, solve for the variable through basic arithmetic operations. Continuing our example, after isolating \( 5x \), divide both sides by 5 to solve for \( x \).
• Check Your Solution: Substitute your solution back into the original equation to ensure it satisfies the equation.

By following these systematic steps, you can solve any linear equation with confidence.

Translating Sentences to Mathematical Equations

Mathematical translation is the process of converting words into mathematical expressions or equations. This is an essential skill in problem-solving, as word problems often describe the relationship between quantities in plain language.

When translating, look for keywords that indicate operations:

• Addition: Keywords like 'sum', 'increased by', or 'more than'.
• Subtraction: Words such as 'difference', 'less than', or 'decreased by'.
• Multiplication: Terms like 'product', 'times', or 'of'.
• Division: Phrases such as 'quotient' or 'divided by'.
• Equality: Words like 'is', 'is equal to', or 'equals'.

For instance, the sentence "The difference of \( 5x \) and 6 is equal to 4" translates to the equation \( 5x - 6 = 4 \). By identifying the key terms "difference" and "is equal to," we can form the correct mathematical statement. Efficient translation is a crucial skill for tackling word problems in algebra.