Problem 86 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{x}{6}+1=4 $$

Step-by-Step Solution

Answer

The solution to the equation \( \frac{x}{6}+1=4 \) is \( x=18 \).

Step 1: Isolate the Variable Term

To solve the equation \( \frac{x}{6} + 1 = 4 \), we first need to isolate the variable term \( \frac{x}{6} \). To do this, subtract 1 from both sides of the equation:\[\frac{x}{6} + 1 - 1 = 4 - 1\]This simplifies to:\[\frac{x}{6} = 3\]

Step 2: Eliminate the Fraction

Now that we have \( \frac{x}{6} = 3 \), we need to eliminate the fraction to solve for \( x \). Multiply both sides of the equation by 6 to get:\[6 \times \frac{x}{6} = 6 \times 3\]The left side simplifies to \( x \) and the right side becomes 18, so:\[x = 18\]

Step 3: Verify the Solution

To ensure that the solution \( x = 18 \) is correct, substitute it back into the original equation:\[\frac{18}{6} + 1 = 4\]This simplifies to:\[3 + 1 = 4\]Since both sides of the equation are equal, the solution is verified as correct.

Key Concepts

Isolate the Variable

To begin solving an equation like \(\frac{x}{6} + 1 = 4\), we must first focus on isolating the variable term. This concept involves getting the variable by itself on one side of the equation. In this instance, the variable is inside the fraction \(\frac{x}{6}\). To isolate it means we need to eliminate other numbers on the same side of the equation as the variable term.

• Start by removing any constants that are added or subtracted alongside the variable term. Here, we have \(+1\), so we subtract 1 from both sides of the equation.
• This equilibration doesn't change the equation because we perform the same operation on both sides, maintaining the equation's balance.

By doing this, we simplify the problem to \(\frac{x}{6} = 3\). Now, the equation is set up to focus solely on the variable term.

Eliminate Fractions

Fractions can make equations look tricky, but the key to simplifying them is through elimination. When you have \(\frac{x}{6} = 3\), you need to remove the fraction to find the value of \(x\). The process typically involves:

• Multiplying both sides of the equation by the denominator of the fraction. This tactic helps to eliminate the fraction and get a clearer, whole number equation.
• In our equation, multiplying both sides by 6 cancels out the denominator, leaving \(x\) by itself on the left side.
• The right side, \(6 \times 3\), simplifies to 18, resulting in \(x = 18\).

This straightforward multiplication ensures that \(x\) is freed from the fraction and makes solving the equation easier.

Verify Solution

After finding a solution, the essential step is to verify it to ensure correctness. Verifying involves substituting the solved value back into the original equation to check if it holds true. Here's how:

• Take your solution \(x = 18\) and substitute it into the place of \(x\) in the original equation \(\frac{x}{6} + 1 = 4\).
• Calculate \(\frac{18}{6} + 1\), which simplifies to \(3 + 1 = 4\).
• This equals the right side of the equation, confirming both sides are the same, which proves that your solution is correct.

Verification provides a confidence check, ensuring no calculation errors have been made and that your solution is indeed valid.