Problem 14 - Algebra A Combined Function

Question

$$ 3 n+2 n=7+4 n $$

Step-by-Step Solution

Answer

The solution is \( n = 7 \).

Step 1: Combine Like Terms

First, let's combine the like terms on the left side of the equation. You have two terms involving \( n \): \( 3n \) and \( 2n \). Adding them together gives us \( 5n \). So, the equation becomes \( 5n = 7 + 4n \).

Step 2: Move Variable Terms to One Side

To isolate \( n \), we need to move all terms involving \( n \) to one side of the equation. Subtract \( 4n \) from both sides of the equation. This gives us \( 5n - 4n = 7 \), which simplifies to \( n = 7 \).

Step 3: Verify Solution

Substitute \( n = 7 \) back into the original equation to verify the solution. The original equation is \( 3n + 2n = 7 + 4n \). Plugging \( 7 \) in gives \( 3(7) + 2(7) = 7 + 4(7) \), which simplifies to \( 21 + 14 = 7 + 28 \). Both sides equal \( 35 \), confirming that \( n = 7 \) is the correct solution.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. In the given exercise, we have an expression with the variable \( n \), specifically:

• \( 3n \)
• \( 2n \)
• \( 4n \)
• Constant term: 7

The presence of these mathematical components allows us to form equations and solve for unknown variables, like \( n \). Understanding how to manipulate these expressions is crucial for solving algebraic equations.

Combining Like Terms

Combining like terms is a fundamental step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. In the equation \( 3n + 2n = 7 + 4n \), the terms \( 3n \) and \( 2n \) are like terms because they both contain the variable \( n \). To combine them, simply add their coefficients (the numbers in front of the variables):

• \( 3n + 2n = 5n \)

This simplifies the equation to \( 5n = 7 + 4n \). Simplifying by combining like terms simplifies your work and brings clarity to solving equations.

Solving Equations

To solve an equation, the main goal is to find the value of the variable that makes the equation true. For the equation \( 5n = 7 + 4n \), follow these steps:1. **Isolate the variable.** We want all terms with \( n \) on one side. Subtract \( 4n \) from both sides:

• \( 5n - 4n = 7 \)

Which simplifies to \( n = 7 \).
2. When the equation has been simplified to something like \( n = 7 \), you have solved the equation. Isolating terms allows clearer understanding and reduces errors.

Verification of Solutions

Verification is the final and crucial step in solving equations. It ensures that the solution you found is correct. Substitute \( n = 7 \) back into the original equation \( 3n + 2n = 7 + 4n \):

• On the left: \( 3(7) + 2(7) = 21 + 14 = 35 \)
• On the right: \( 7 + 4(7) = 7 + 28 = 35 \)

Both sides equal 35, confirming that \( n = 7 \) makes the original equation true.Checking your solution prevents mistakes and gives confidence in your problem-solving process. Verification is especially helpful when solving more complex equations.