8-1/4 X equals 2/5 X +14 what would this answer be in solving equations by clearing the fractions?

Answer 1

To solve the equation \( 8 - \frac{1}{4} X = \frac{2}{5} X + 14 \) by clearing the fractions, follow these steps:

1. Identify the denominators: The denominators present in the equation are 4 and 5.

2. Find the least common multiple (LCM): The LCM of 4 and 5 is 20.

3. Multiply the entire equation by 20 to eliminate the fractions:

\[ 20(8 - \frac{1}{4} X) = 20(\frac{2}{5} X + 14) \]

4. Distribute 20 across both sides:

\[ 20 \cdot 8 - 20 \cdot \frac{1}{4} X = 20 \cdot \frac{2}{5} X + 20 \cdot 14 \]

5. Calculate each term:

• For the left side:

\[ 20 \cdot 8 = 160 \] \[ 20 \cdot \frac{1}{4} X = 5X \quad (\text{because } 20 \div 4 = 5) \] So, the left side becomes: \[ 160 - 5X \]

• For the right side:

\[ 20 \cdot \frac{2}{5} X = 8X \quad (\text{because } 20 \div 5 \cdot 2 = 8) \] \[ 20 \cdot 14 = 280 \] So, the right side becomes: \[ 8X + 280 \]

6. Rewrite the equation without fractions:

\[ 160 - 5X = 8X + 280 \]

7. Rearrange the equation to isolate \(X\). Add \(5X\) to both sides:

\[ 160 = 8X + 5X + 280 \] \[ 160 = 13X + 280 \]

8. Subtract 280 from both sides:

\[ 160 - 280 = 13X \] \[ -120 = 13X \]

9. Divide both sides by 13:

\[ X = -\frac{120}{13} \]

So, the solution to the equation is:

\[ X = -\frac{120}{13} \approx -9.23 \]

This is the solution after clearing the fractions.