Problem 6 - Algebra for College Students

Question

Solve each equation. \(n-0.5 n=12\)

Step-by-Step Solution

Answer

The solution is \(n = 24\).

Step 1: Simplify the Left Side

First, notice that the left side of the equation, \(n - 0.5n = 12\), can be combined. Since \(n\) is the same as \(1n\), subtracting \(0.5n\) from \(1n\) results in \(0.5n\). Therefore, the equation simplifies to \(0.5n = 12\).

Step 2: Solve for n

To solve for \(n\), divide both sides of the equation by 0.5. This gives \(n = \frac{12}{0.5}\).

Step 3: Calculate the Value

Calculate \(\frac{12}{0.5}\) by multiplying 12 by 2 (since dividing by 0.5 is equivalent to multiplying by 2). This results in \(n = 24\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial step in solving equations efficiently. In our example, the equation is \( n - 0.5n = 12 \). The expression on the left consists of two terms that involve \( n \). It's important to recognize that \( n \) is the same as \( 1n \). This understanding allows us to combine like terms.

• \( n \) or \( 1n \) represents one whole of \( n \).
• Subtracting \( 0.5n \) is straightforward: simply subtract the coefficients.
• Thus, \( 1n - 0.5n = 0.5n \).

By simplifying the left side, our equation becomes \( 0.5n = 12 \), making it easier to solve. Recognizing and combining like terms helps to simplify the equation, avoiding more complex steps down the line.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. Once we've simplified our equation to \( 0.5n = 12 \), we can solve for \( n \) by isolating it on one side. Here, \( 0.5n \) means \( n \) is being multiplied by 0.5. To solve an equation, reverse the operations around the variable:

• In this case, to isolate \( n \), divide both sides by 0.5.
• Dividing both sides of an equation by the same number keeps the equation balanced.

After dividing both sides by 0.5, the equation becomes \( n = \frac{12}{0.5} \). The variable \( n \) is now isolated, and you can calculate its value.

Division in Algebra

Division in algebra can sometimes involve fractions or decimals, as seen with the division step in solving \( 0.5n = 12 \). Here's a simple way to understand it:When dividing by a fraction or a decimal, consider how to express it as multiplication:

• Dividing by 0.5 is the same as multiplying by 2, because 0.5 is the reciprocal of 2 (\( \frac{1}{0.5} = 2 \)).
• Think of it as determining how many times 0.5 fits into 12, or equivalently, how many halves make up 12.

By recalculating \( \frac{12}{0.5} \) as \( 12 \times 2 \), you find that \( n = 24 \). This method showcases how division by a decimal is simplified to multiplication, making the operation easier to compute.