Problem 14 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 8 x=81 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = \frac{81}{8} \).

Step 1: Isolate the Variable

To solve the equation \( 8x = 81 \), we need to isolate \( x \) on one side of the equation. We do this by dividing both sides by 8, which the coefficient of \( x \). This gives us \( x = \frac{81}{8} \).

Step 2: Simplify the Fraction

The next step is to simplify \( x = \frac{81}{8} \) if possible. In this case, \( 81 \) and \( 8 \) do not have common factors other than 1, so the fraction is already in its simplest form. Therefore, \( x = \frac{81}{8} \).

Step 3: Check the Solution

To ensure our solution is correct, substitute \( x = \frac{81}{8} \) back into the original equation. Calculate \( 8 \times \frac{81}{8} \). Since \( 8 \) in the numerator cancels out with the \( 8 \) in the denominator, we are left with \( 81 \). Thus, the left side equals the right side: \( 81 = 81 \). Therefore, \( x = \frac{81}{8} \) is the correct solution.

Key Concepts

Isolating Variables

When solving linear equations, the primary goal is to isolate the variable on one side of the equation to determine its value. This process involves reversing the mathematical operations surrounding the variable. In the equation \( 8x = 81 \), the variable \( x \) is being multiplied by 8. To isolate \( x \), we perform the inverse operation, which is division.

• Divide both sides by the same number: In this case, divide 81 by 8 to cancel out the multiplication by 8 on the left side.
• The resulting equation, after dividing both sides by 8, is \( x = \frac{81}{8} \).

This step is crucial because it transforms the equation into one where \( x \) stands alone, making it easier to find its exact value. Remember, whatever operation you perform on one side must be done to the other to maintain equality.

Simplifying Fractions

Once the variable is isolated, we often need to simplify the resulting fraction to its simplest form. Simplifying involves reducing the fraction by dividing the numerator and denominator by their greatest common divisor (GCD). For \( \frac{81}{8} \), the simplification process checks if 81 and 8 have any common factors other than 1.

• In this case, since 81 and 8 share no common factors other than 1, the fraction \( \frac{81}{8} \) is already fully simplified.
• This means there are no smaller integers that divide both 81 and 8 evenly.

Understanding simplification ensures that the solution is in its most concise form, making it easier to work with or interpret in further mathematical processes. Always strive to present your final answer in its simplest form unless specified otherwise.

Solution Verification

To verify the correctness of a solution in mathematics, we substitute the derived value back into the original equation. This process confirms that the value satisfies the equation's conditions. Checking the solution for the equation \( 8x = 81 \) involves replacing \( x \) with \( \frac{81}{8} \).

• Substitute: Calculate \( 8 \times \frac{81}{8} \).
• Cancel the multiplied terms: The 8 in the numerator cancels with the 8 in the denominator.
• Final result: This leaves you with 81, which is equal to the original right side of the equation.

When both sides of the equation are equal, the calculated value is confirmed as correct. This simple verification step is essential to ensure no errors occurred during isolation or simplification.