Problem 46 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ \frac{8 k}{3}=32 $$

Step-by-Step Solution

Answer

The solution is \( k = 12 \).

Step 1: Isolate the Variable

We start with the equation \( \frac{8k}{3} = 32 \). To isolate \( k \), first multiply both sides by 3 to get rid of the fraction: \( 8k = 96 \).

Step 2: Solve for the Variable

Now we need to solve for \( k \). Since \( 8k = 96 \), divide both sides by 8 to find \( k \). Thus, \( k = \frac{96}{8} = 12 \).

Step 3: Check the Solution

Substitute \( k = 12 \) back into the original equation to verify the solution. Substitute and simplify: \( \frac{8 \times 12}{3} = 32 \). Simplifying this gives \( \frac{96}{3} = 32 \), which checks out as \( 32 = 32 \). This confirms \( k = 12 \) is the correct solution.

Key Concepts

Solving for a Variable

When you encounter a linear equation like \( \frac{8k}{3} = 32 \), the goal is to find the value of the variable, which in this case is \( k \). This process of solving for a variable involves performing algebraic operations to determine what \( k \) equals. To solve for \( k \), you must manipulate the equation so that \( k \) stands alone on one side of the equation. This requires systematic steps:

• Identify the operation that affects the variable and reverse it.
• Perform the same operation on both sides to maintain the equation's balance.

For instance, if \( k \) is multiplied by a number (as it is here with 8), you'll eventually divide by that number. The principle is to "undo" operations to isolate the variable, leading to the solution.

Isolate the Variable

To isolate the variable means to get the variable by itself on one side of the equation. In our equation \( \frac{8k}{3} = 32 \), we need to eliminate the fraction so that \( k \) can be easily expressed. Here's how you can isolate \( k \):

• Remove the fraction: Multiply both sides of the equation by 3 to remove the denominator, resulting in \( 8k = 96 \).
• Solve for the variable: Once the fraction is removed, you focus on the remaining multiplication. By dividing both sides by 8, you can solve for \( k \), resulting in \( k = \frac{96}{8} \).

These steps ensure each operation is systematically reversed, isolating \( k \) effectively.

Check the Solution

After finding a potential solution for the variable, it's essential to verify its correctness by substituting it back into the original equation. This step ensures that no errors were made during calculations.For \( k = 12 \), substitute into the original equation \( \frac{8 \times 12}{3} \):

• Calculate the numerator: \( 8 \times 12 = 96 \).
• Divide by 3: \( \frac{96}{3} = 32 \).
• Verify equality: Confirm that \( 32 \) equals the right-hand side of the original equation.

This process, called substitution, confirms that our value for \( k \) satisfies the equation because the left side equals the right side, assuring the solution's accuracy.