Problem 9 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 1 through \(6 .\) \(\frac{1}{6} d=\frac{1}{2}\)

Step-by-Step Solution

Answer

The solution is \( d = 3 \).

Step 1: Understand the Equation

The equation presented is \( \frac{1}{6} d = \frac{1}{2} \), which means one-sixth of \( d \) is equal to one-half. Our goal is to find the value of \( d \).

Step 2: Eliminate the Fraction

To simplify, we'll eliminate the fraction by multiplying both sides of the equation by 6. This gives us: \( 6 \times \frac{1}{6} d = 6 \times \frac{1}{2} \).

Step 3: Simplify the Equation

Simplifying the equation from Step 2 results in \( d = 3 \). This is because \( 6 \times \frac{1}{6} = 1 \) on the left, and \( 6 \times \frac{1}{2} = 3 \) on the right.

Step 4: Verify the Solution

To verify, substitute \( d = 3 \) back into the original equation \( \frac{1}{6} d = \frac{1}{2} \). This gives us \( \frac{1}{6} \times 3 = \frac{1}{2} \). Simplifying, \( \frac{3}{6} = \frac{1}{2} \) confirms the solution is correct.

Key Concepts

Eliminating Fractions

Fractions in equations can make solving them a bit daunting at first glance. Therefore, one effective method is eliminating them, simplifying the equation to make it easier to find the solution. Such a method involves transferring the equation into a form without fractions.

To eliminate a fraction, identify the common denominator of the fractions. Once identified, multiply each part of the equation by this number. This step works by effectively "canceling out" the denominators, which makes the fractions disappear.

• For instance, in the equation \(\frac{1}{6}d = \frac{1}{2}\), the denominator is 6.
• Multiply every term by 6, which means applying 6 to both sides: \(6 \times \frac{1}{6}d = 6 \times \frac{1}{2}\).

After simplifying, this results in \(d = 3\). Multiplying both sides ensures the equation remains balanced, and greatly simplifies the path to finding the solution.

Equation Verification

After solving an equation, it's crucial to verify your solution to confirm its accuracy. Verification is the process where we re-substitute the found value back into the original equation. This not only checks if you've got the right answer, but also reinforces understanding of the problem.

Let's verify the solution by substituting it back:

• Take the solution \(d = 3\) and replace \(d\) in the original equation \(\frac{1}{6}d = \frac{1}{2}\).
• This gives us \(\frac{1}{6} \times 3 = \frac{1}{2}\).

Upon simplifying, \(\frac{3}{6} = \frac{1}{2}\), which is indeed true, confirming our solution is correct. Verification ensures that every step was correctly executed and adds confidence to your math skills.

Substitution Method

The substitution method is a valuable technique in algebra that involves replacing a variable with a given or previously determined equivalent value. Although it is commonly used in systems of equations, it also plays a role in verifying solutions in simple equations.

This method takes the expression obtained from solving and substitutes it back instantly into the initial equation to see if both sides are equal.

• For example, in the exercise, \(d = 3\) was found by simplifying the expression after eliminating fractions.
• By substituting \(d = 3\) into the original expression \(\frac{1}{6}d = \frac{1}{2}\), it becomes \(\frac{1}{6} \times 3 = \frac{1}{2}\).

The values equate after simplification, illustrating the result is accurate. Substitution is key in checking the validity of solutions, offering verification through re-engagement with the original problem.