Problem 38 - Prealgebra

Question

Solve each equation by finding a number to replace \(n\) that will make the equation a true statement. $$8 \cdot n=16$$

Step-by-Step Solution

Answer

The value of \(n\) is 2.

Step 1: Understand the Equation

The given equation is \(8 \cdot n = 16\). This means 8 multiplied by some number \(n\) equals 16. Our goal is to find the value of \(n\) that makes this equation true.

Step 2: Isolate the Variable

To solve for \(n\), we need to isolate it on one side of the equation. Currently, \(n\) is being multiplied by 8. The opposite operation of multiplication is division, so we will divide both sides by 8 to isolate \(n\).

Step 3: Perform Division

Divide both sides of the equation by 8 to find \(n\):\[ n = \frac{16}{8} \]

Step 4: Simplify the Division

Calculate the division on the right side of the equation:\[ n = 2 \]

Step 5: Verify the Solution

To ensure our solution is correct, substitute \(n = 2\) back into the original equation:\(8 \cdot 2 = 16\).Since the left side equals the right side (16 = 16), our solution is verified.

Key Concepts

Multiplication Equations

A multiplication equation involves a variable and a numerical coefficient that are multiplied together. In our example, the equation is given as \(8 \cdot n = 16\). This tells us that the number 8 is multiplied by some unknown number, denoted as \(n\), and the product is 16. Solving multiplication equations consists of finding the unknown variable that turns the equation into a true statement.

To solve multiplication equations effectively, remember:

• The coefficients and variables are always multiplied together.
• You are looking for a number that, when multiplied by the coefficient, results in the value on the other side of the equation.

When solving, understanding this structure is vital to finding the value for the variable that satisfies the equation.
Keep practicing, as multiplication equations form the foundation of many algebraic problems you'll encounter later.

Variable Isolation

Variable isolation is a fundamental concept in algebra, where the aim is to get the variable on its own on one side of the equation. In the equation \(8 \cdot n = 16\), we need to isolate \(n\). Currently, it is teamed up with 8 through multiplication.

To isolate \(n\):

• Identify the operation being performed on the variable: here it's multiplication by 8.
• Apply the inverse operation to both sides of the equation. Since we're dealing with multiplication, we use division.

By dividing both sides of the equation by 8, we simplify the process:
\[ n = \frac{16}{8} \]
This results in \(n\), standing alone with its solution on one side. Variable isolation sets the stage for solving the equation easily, allowing you to unwrap complex algebra tasks as they arise.

Verification of Solutions

Verification of solutions ensures that the obtained solution satisfies the original equation. After isolating and calculating \(n\) in the example \(8 \cdot n = 16\), we found that \(n = 2\).

To verify the solution:

• Substitute the calculated value back into the original equation.
• Check if both sides of the equation are equal with this substitution.

Substituting \(n = 2\) back, we get:
\(8 \cdot 2 = 16\)
Since 16 equals 16 on both sides, the solution holds true.
Verification is a crucial step as it confirms that the solution is correct and consistent with the original equation conditions.
Always perform verification; it's a reliable check to avoid mistakes in algebra.