Problem 9 - Prealgebra

Question

Solve each equation. Check your solution. $$3 n+4=3(n+2)$$

Step-by-Step Solution

Answer

The equation has no solution.

Step 1: Expand the Equation

Expand the right side of the equation: \(3(n+2)\).This becomes \(3n + 6\). Thus, the equation now is:\[3n + 4 = 3n + 6\]

Step 2: Eliminate Like Terms

Subtract \(3n\) from both sides of the equation to eliminate the \(3n\) terms:\[3n + 4 - 3n = 3n + 6 - 3n\]This simplifies to:\[4 = 6\]

Step 3: Analyze the Result

The resulting equation \(4 = 6\) is a contradiction. This means that no value of \(n\) can satisfy the original equation.

Step 4: Conclude the Solution

Since the steps above led to a contradiction, we conclude that the equation has no solution. There is no value of \(n\) that makes the equation true.

Key Concepts

Contradiction in Equations

Equations are used to find unknown values but can sometimes lead to contradictions, which occur when two sides of an equation cannot be equal, no matter the value of the variable. In the given exercise, solving the equation \(3n + 4 = 3(n + 2)\) eventually leads to an impossible statement: \(4 = 6\). A contradiction means the equation has no solution.

Recognizing contradictions is essential as it indicates that the equation is either incorrectly set up or inherently unsolvable. When an equation like this one emerges from simplifying expressions, it is vital to examine the original set-up or the logic behind the problem.

Always double-check your work when you find contradictions. Ensure there were no errors in algebraic manipulations or steps. However, if your work checks out, then it confirms the absence of a solution.

Prealgebra Concepts

Understanding prealgebra concepts is critical for solving equations effectively. These concepts include working with basic operations, distributive property, and understanding variables.

• Basic Operations: Addition, subtraction, multiplication, and division are the foundational tools used to manipulate equations.
• The Distributive Property: Applied in transforming expressions, such as distributing a number across terms inside parentheses. Example: \(3(n+2)\) becomes \(3n + 6\).
• Variables: Letters that represent unknowns. In this exercise, \(n\) is the variable we are solving for.

A firm grasp of these concepts allows you to simplify and solve equations step-by-step. Remember, the goal is to isolate the variable on one side of the equation to find its value.

Practicing these skills regularly helps in identifying patterns and quickly spotting contradictions in equations.

Step-by-Step Problem Solving

Step-by-step problem solving involves breaking down complex problems into manageable components, allowing for a clearer understanding and systematic approach.

• Step 1: Identify and Distribute: Expand any terms involving parentheses, as it helps in simplifying the equation.
• Step 2: Simplify and Eliminate Like Terms: Focus on combining similar terms to ease solving. This provides a clearer path to isolate the variable.
• Step 3: Analyze the Results: Once simplified, evaluate the equation. If the result is nonsensical like \(4 = 6\), a contradiction is present.
• Step 4: Draw Conclusions: Determine the nature of the solution. Conclude whether the equation has a solution or not.

Following this method ensures a thoughtful and thorough approach to solving arithmetic problems, especially ones involving potential contradictions. Regular practice using step-by-step methods encourages accuracy and reduces mistakes.