Problem 17 - Prealgebra

Question

Solve each equation. Check your solution. $$2 n-5=21$$

Step-by-Step Solution

Answer

The solution is \(n = 13\).

Step 1: Isolate the variable

To solve the equation \(2n - 5 = 21\), we need to start by isolating the variable \(n\). First, add 5 to both sides of the equation to remove the -5 from the left side. This gives us: \[2n - 5 + 5 = 21 + 5\]Simplifying both sides, we get:\[2n = 26\]

Step 2: Solve for the variable

Now that we have \(2n = 26\), we need to solve for \(n\) by dividing both sides by 2. This isolates the variable \(n\):\[\frac{2n}{2} = \frac{26}{2}\]Simplifying this gives us:\[n = 13\]

Step 3: Check the solution

To ensure our solution \(n = 13\) is correct, we substitute it back into the original equation to verify:\[2(13) - 5 = 21\]Calculating inside the parentheses first: \[2 \times 13 = 26\]Then subtract 5 from 26:\[26 - 5 = 21\]Since both sides of the equation are equal, our solution is verified.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving equations. It involves rearranging and simplifying expressions to make solving them easier. When working with the equation \(2n - 5 = 21\), we begin by addressing the manipulation of terms. Our goal is to maintain the balance of the equation while altering its appearance.

Key operations in algebraic manipulation include:

• Adding or Subtracting Terms: Use these operations to eliminate numbers or terms from one side of the equation.
• Multiplying or Dividing Terms: Apply these operations when trying to simplify coefficients of the variable.

In our exercise, we start by adding 5 to both sides. This cancels out the \(-5\) on the left side and transitions the equation into a simpler form. Through consistent and strategic algebraic manipulation, we transform the original equation into one where it's easier to solve for \(n\).

Remember, each step in manipulation should aim to get closer to isolating the variable, maintaining the equation's integrity.

Equation Verification

Verification is a crucial part of solving any equation. After finding a solution, checking its validity ensures that no mistakes were made during the algebraic manipulation and calculation steps. This step involves substituting the obtained result back into the original equation.

In our example, we solved for \(n = 13\). To verify, we replace \(n\) in the original equation with 13. Specifically, this means computing:

• \(2 \times 13 - 5\)
• This simplifies to \(26 - 5\)
• Finally, comparing this to the other side of the original equation, \(21\)

Both sides equal 21, confirming the correctness of our solution.

Verification acts as a safety net. By cross-referencing the solution with the original problem, we confirm its accuracy. Always ensure your calculated solution satisfies the initial conditions of the equation.

Isolating Variables

Isolating the variable is fundamental in solving linear equations. By focusing on one variable, we clarify the equation's structure, making it easier to solve. Typically, this process involves reversing the operations affecting the variable.

Key steps when isolating variables include:

• Identifying the variable term: In \(2n - 5 = 21\), it’s \(2n\).
• Removing surrounding numbers or terms: Initially, we add 5 to eliminate the \(-5\) from \(2n\).
• Solving for the variable: Conclude by dividing to free \(n\) from its coefficient \((2)\).

Through these steps, isolating the variable \(n\) transforms the problem from managing many terms to just finding one number. Our goal is to simplify the equation until it clearly expresses the variable in terms of known quantities. Successfully doing this simplifies solving and verifying the equation.