Problem 29 - Prealgebra

Question

Solve equation. Check your solution. \(4.3 n-1.6=2.3 n+5.2\)

Step-by-Step Solution

Answer

The solution is \(n = 3.4\).

Step 1: Move Variable Terms Together

First, subtract \(2.3n\) from both sides of the equation to get the variable terms on one side: \(4.3n - 2.3n - 1.6 = 5.2\).

Step 2: Simplify the Equation

Simplify the left side by combining like terms: \(2.0n - 1.6 = 5.2\).

Step 3: Isolate the Variable Term

Add \(1.6\) to both sides to isolate the term with the variable: \(2.0n = 5.2 + 1.6\).

Step 4: Simplify the Right Side

Simplify the right side by performing the addition: \(2.0n = 6.8\).

Step 5: Solve for the Variable

Divide both sides by \(2.0\) to solve for \(n\): \(n = \frac{6.8}{2.0}\).

Step 6: Simplify the Solution

Calculate the division: \(n = 3.4\).

Step 7: Check the Solution

Substitute \(n = 3.4\) back into the original equation to verify: \(4.3(3.4) - 1.6 = 2.3(3.4) + 5.2\)Calculate both sides to check for equality: Left side: \(4.3 \times 3.4 = 14.62\)\(14.62 - 1.6 = 13.02\)Right side: \(2.3 \times 3.4 = 7.82\)\(7.82 + 5.2 = 13.02\)Both sides are equal, confirming \(n = 3.4\) is correct.

Key Concepts

Prealgebra Simplified

Prealgebra is the foundation for understanding algebraic concepts. It is the beginning of working with variables, numbers, and arithmetic operations to solve equations.
It typically involves the introduction of expressions and equations, similar to what you encounter in the given exercise.
The goal is to find the value of the variable, in this case, represented by \(n\). Understanding prealgebra lays the groundwork for developing problem-solving skills that are crucial for higher-level math.
Key elements of prealgebra include:

• Understanding variables: A variable, like \(n\), is a symbol that represents an unknown number.
• Equations: An equation is a mathematical statement showing that two expressions are equal.
• Basic operations: Knowing when to use addition, subtraction, multiplication, and division.

Mastering Checking Solutions

After solving an equation, it's essential to verify the solution you have found. Checking the solution ensures accuracy and confirms that no mistakes have been made during the calculation process.
This involves substituting the solution back into the original equation to ensure that both sides of the equation are equal.
For example, if you find \( n = 3.4 \), substitute \(3.4\) back into the original equation:

• Left side: Calculate \(4.3 \times 3.4 - 1.6\).
• Right side: Calculate \(2.3 \times 3.4 + 5.2\).
• Verify equality: Both sides should result in the same value (\(13.02\)), confirming that \(n = 3.4\) is correct.

Checking solutions is a valuable practice, not only to validate your work but also to strengthen your understanding of the solution process.

The Step-by-Step Solution Approach

Breaking down a problem into manageable steps makes complex equations easier to solve. Let's walk through the systematic process used in the solution:
1. **Move Variable Terms Together**: Start by moving all terms with the variable to one side of the equation.
This is done by subtracting \(2.3n\) from both sides.
2. **Simplify the Equation**: Combine the like terms on each side to simplify. Here, \(4.3n - 2.3n\) becomes \(2.0n\).
3. **Isolate the Variable**: Remove any constants from the side with the variable. In this example, add \(1.6\) to both sides to isolate \(2.0n\).
4. **Solve for the Variable**: Once isolated, divide to solve for the variable.
In this exercise, divide \(6.8\) by \(2.0\) to find \(n = 3.4\).

• Moving step by step clarifies the procedure, allowing for a clear understanding of each action's purpose.
• This approach helps ensure accuracy by highlighting each calculation and transformation.

The Art of Combining Like Terms

Combining like terms is a fundamental skill in solving equations. It involves simplifying expressions by adding or subtracting terms that have the same variables raised to the same power.
In our example, like terms are combined:

• \(4.3n\) and \(2.3n\) are both terms involving \(n\). Subtracting them gives \(2.0n\).
• Constant terms on each side are similarly handled: when you add \(1.6\) to both sides, you simplify the constant terms.

Combining like terms streamlines the equation, making it much easier to solve. This step is crucial as it directly affects the simplicity and solvability of the equation. Mastery of this concept allows for quick transformations and straightforward paths to solutions.