Solving linear equations with two fractions - Variation Theory - Free Printable

Problem Analysis:

We are tasked with solving the equation:
\[
\frac{x + 2}{6} - \frac{x - 4}{8} = 9
\]

Step-by-Step Solution:

#### Step 1: Write down the given equation
\[
\frac{x + 2}{6} - \frac{x - 4}{8} = 9
\]

#### Step 2: Find a common denominator
The denominators are 6 and 8. The least common multiple (LCM) of 6 and 8 is 24. We will rewrite each fraction with a denominator of 24.

- For \(\frac{x + 2}{6}\):
\[
\frac{x + 2}{6} = \frac{4(x + 2)}{24} = \frac{4x + 8}{24}
\]

- For \(\frac{x - 4}{8}\):
\[
\frac{x - 4}{8} = \frac{3(x - 4)}{24} = \frac{3x - 12}{24}
\]

#### Step 3: Rewrite the equation with the common denominator
Substitute the rewritten fractions back into the equation:
\[
\frac{4x + 8}{24} - \frac{3x - 12}{24} = 9
\]

Since the denominators are the same, combine the numerators:
\[
\frac{(4x + 8) - (3x - 12)}{24} = 9
\]

Simplify the numerator:
\[
(4x + 8) - (3x - 12) = 4x + 8 - 3x + 12 = x + 20
\]

So the equation becomes:
\[
\frac{x + 20}{24} = 9
\]

#### Step 4: Eliminate the denominator
Multiply both sides of the equation by 24 to clear the denominator:
\[
x + 20 = 9 \cdot 24
\]

Calculate \(9 \cdot 24\):
\[
9 \cdot 24 = 216
\]

So the equation is:
\[
x + 20 = 216
\]

#### Step 5: Solve for \(x\)
Subtract 20 from both sides:
\[
x = 216 - 20
\]
\[
x = 196
\]

Final Answer:

\[
\boxed{196}
\]

Verification:

To verify, substitute \(x = 196\) back into the original equation:
\[
\frac{x + 2}{6} - \frac{x - 4}{8} = 9
\]

Substitute \(x = 196\):
\[
\frac{196 + 2}{6} - \frac{196 - 4}{8} = 9
\]

Simplify each term:
\[
\frac{198}{6} - \frac{192}{8} = 9
\]

Calculate each fraction:
\[
\frac{198}{6} = 33 \quad \text{and} \quad \frac{192}{8} = 24
\]

So the equation becomes:
\[
33 - 24 = 9
\]

This is true, confirming that our solution is correct.

\[
\boxed{196}
\]