Problem 19 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ -5(n-2)=8-4 n $$

Step-by-Step Solution

Answer

The solution is n = 2.

Step 1: Distribute on the Left Side

Distribute the -5 across the terms inside the parenthesis (n-2) . This means you multiply -5 by n and -2 , which gives you -5n + 10 . Thus, the equation becomes -5n + 10 = 8 - 4n .

Step 2: Move Variables to One Side

Add 4n to both sides to get all the n terms on one side of the equation. This results in -5n + 4n + 10 = 8 , which simplifies to -n + 10 = 8 .

Step 3: Isolate the Variable

Subtract 10 from both sides to isolate the variable term -n . This yields -n = 8 - 10 , which simplifies to -n = -2 .

Step 4: Solve for the Variable

To solve for n , multiply both sides of the equation -n = -2 by -1 . This results in n = 2 .

Step 5: Verify the Solution

Substitute n = 2 back into the original equation -5(n - 2) = 8 - 4n to verify the solution. Simplifying both sides gives: -5(2 - 2) = 8 - 4(2) , which becomes 0 = 0 , confirming the solution is correct.

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that simplifies expressions and equations by distributing a factor over addition or subtraction inside parentheses. In the exercise example, we encounter the expression \(-5(n-2)\). This means that \(-5\) is distributed across both \(n\) and \(-2\).

This gives us:\[-5 \cdot n + (-5) \cdot (-2) = -5n + 10\]

So, it turns a single expression with parentheses into two separate terms without parentheses. By applying the distributive property, we simplify one side of the equation, preparing it for further steps like combining like terms and isolating variables.

Isolation of Variables for Solving

Isolating variables is a critical step in solving linear equations as it focuses on getting the unknown, often represented by a variable like \(n\), by itself on one side of the equation. This makes it easier to determine the value of the variable.

In the original exercise, after applying the distributive property, we have the equation \(-5n + 10 = 8 - 4n\). The goal is to rearrange it so that \(n\) is isolated. We do this by:

• Adding \(4n\) to both sides to collect all \(n\) terms on one side: \(-5n + 4n\) becomes \(-n\).
  
  
• Subtracting \(10\) from both sides so that constants are removed from the side of the unknown: \(-n + 10 = 8\) simplifies to \(-n = -2\).

This strategic manipulation prepares the equation for the final solving step, making \(n\) easy to solve.

Verifying Solutions to Ensure Accuracy

Ensuring that the obtained solution is correct is an indispensable part of solving equations. It involves substituting the value of the unknown back into the original equation to see if a true statement results.

In our exercise, after finding \(n = 2\), verification is done by substituting \(n\) back into the original equation \(-5(n - 2) = 8 - 4n\). By performing the calculations:

• On the left side: \[-5(2 - 2) = -5 \cdot 0 = 0\]
  
  
• On the right side: \[8 - 4 \times 2 = 8 - 8 = 0\]

Since both sides equal, \(0 = 0\), the solution \(n = 2\) is correct. This step confirms the validity of our solution, ensuring no errors were made in the calculations or transformations.

Algebraic Simplification Techniques

Algebraic simplification is the process of reducing an algebraic expression to its simplest form. This can involve combining like terms, reducing unnecessary complexity, and factoring where applicable.

In our original problem, several simplification techniques are applied:

• First, the distributive property is used to eliminate parentheses and combine terms: \(-5n + 10\).
• Next, like terms are combined, such as \(-5n + 4n = -n\).
• Finally, constants are simplified: in \(-n + 10 = 8\), subtract \(10\) from both sides to get \(-n = -2\).

Through simplification, the equation becomes much easier to handle, progressively transforming it into a form where finding \(n\) is straightforward, showing the power and importance of simplifying expressions.