Problem 79 - Elementary Algebra

Question

Translate the following sentences into linear equations and then solve. Four added to three times a number \(n\) is \(25 .\)

Step-by-Step Solution

Answer

The value of \(n\) is 7.

Step 1: Identify the Unknown

The problem states that three times a number plus four equals 25. Let's identify our unknown number as the variable \(n\).

Step 2: Translate the Sentence into an Equation

The problem can be expressed as: "Four added to three times a number \(n\) is 25." Translating this into a mathematical equation gives us: \(3n + 4 = 25\).

Step 3: Isolate the Variable

To solve for \(n\), we need to isolate it on one side of the equation. Start by subtracting 4 from both sides of \(3n + 4 = 25\) to get: \(3n = 21\).

Step 4: Solve for the Variable

Divide both sides of \(3n = 21\) by 3 to solve for \(n\): \[ n = \frac{21}{3} = 7. \]

Key Concepts

Variables

In the world of mathematics, variables play a crucial role. A variable is a symbol, often a letter, that represents an unknown value. For example, in our exercise, the letter \( n \) is used as a variable to denote an unknown number.

• Variables are essential because they allow us to create general formulas and expressions that can be applied to various situations.
• They serve as placeholders for values that can change or values we want to find.

Understanding variables and how they work is fundamental in solving equations and working through word problems. A variable embodies versatility and abstraction that is powerful in modeling real-world scenarios mathematically.

Solving Equations

Once we have set up an equation using variables, the next step is solving it. Solving an equation involves finding the value of the unknown variable that makes the equation true.

• To solve the equation \(3n + 4 = 25\), our goal is to get \( n \) by itself on one side of the equation.
• This often involves performing inverse operations, which are operations that reverse the effect of the original operation.

In our example, we start by subtracting 4 from both sides, which cancels out the +4 associated with \( n \). Now, we have \( 3n = 21 \). The next step involves dividing both sides of the equation by 3, the coefficient of \( n \), to solve for \( n \). This gives us \( n = 7 \), which is the solution.
Solving equations is a key skill in algebra that allows us to find unknown values and is a step-by-step process. It’s important to perform the same operation on both sides of an equation to maintain equality.

Mathematical Translation

The process of translating words into mathematical equations, known as mathematical translation, is crucial for solving word problems. Translating involves recognizing key phrases and understanding how they correspond to mathematical operations and expressions.

• In the sentence "Four added to three times a number \( n \) is 25," the phrase "three times" indicates multiplication, and "added to" signifies addition.
• Such signals guide us in forming the equation \(3n + 4 = 25\).

This skill is about converting real-world situations into a language that mathematics can understand and solve. Mathematical translation requires careful reading and interpretation of statements, ensuring that the resulting equation accurately reflects the given scenario. Once properly translated into a linear equation, as in our problem, we've paved the way for solution using algebraic methods.