Problem 23 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 2. $$ \frac{x}{4}-6=1 $$

Step-by-Step Solution

Answer

The solution is \( x = 28 \).

Step 1: Add 6 to Both Sides

To isolate the variable term, add 6 to both sides of the equation: \[ \frac{x}{4} - 6 + 6 = 1 + 6 \] Simplifying this gives: \[ \frac{x}{4} = 7 \]

Step 2: Multiply Both Sides by 4

To eliminate the fraction, multiply both sides of the equation by 4: \[ 4 \times \frac{x}{4} = 7 \times 4 \] This simplifies to: \[ x = 28 \]

Step 3: Check the Solution

Substitute \( x = 28 \) back into the original equation to verify:\[ \frac{28}{4} - 6 = 1 \] Simplifying the left side: \[ 7 - 6 = 1 \] This simplifies to \( 1 = 1 \), confirming our solution is correct.

Key Concepts

Checking Solutions

After solving a linear equation like \( \frac{x}{4} - 6 = 1 \), it's crucial to check if your solution makes the equation true. This step ensures accuracy and helps build confidence in your math skills. Let's see how to verify the result of \( x = 28 \).
To do this, simply substitute the solution back into the original equation. Replace \( x \) with \( 28 \):

• Calculate \( \frac{28}{4} \) which results in \( 7 \).
• Subtract \( 6 \) from \( 7 \), which simplifies to \( 1 \).

Since both sides are equal \( 1 = 1 \), this means \( x = 28 \) is indeed the correct solution. This process of checking is essential, especially when working with more complex equations.

Fraction Elimination

Eliminating fractions in equations can simplify the solving process significantly. Consider our equation \( \frac{x}{4} - 6 = 1 \).
Fractions can be tricky, but with the right approach, they can be easy to handle. Here, multiplying both sides by the denominator of the fraction helps to remove it:

• Multiply every term by \( 4 \) (the denominator) to clear the fraction: \( 4 \left( \frac{x}{4} \right) \).
• This simplifies directly to \( x \) since \( 4 \times \frac{1}{4} = 1 \).

Now, you have a much simpler equation to work with: \( x = 28 \). This approach is very handy for equations containing fractions and speeds up the process of solving them.

Isolation of Variables

Isolating the variable is often the first major step in solving an equation. In our case \( \frac{x}{4} - 6 = 1 \), we want to get the equation to say \( x = \ldots \).
Here's how you isolate the variable:

• First, move all terms not involving the variable to the other side. Add \( 6 \) to both sides to cancel out the \( -6 \): \( \frac{x}{4} = 7 \).
• After clearing the fraction, multiply both sides by \( 4 \) to solve for \( x \).

This step-by-step method makes sure the equation is as simple as possible, and you can see clearly how the solution is derived. Isolation of the variable converts a complex-looking equation into a standard form that is easier to solve.