Questions: Solve the equation. (x+6)/6=1-(x+2)/7

Solution Steps

To solve the given equation, we first eliminate the fractions by finding a common denominator. Then, we simplify the equation and solve for \( x \) by isolating it on one side of the equation.

To eliminate the fractions in the equation

\[ \frac{x+6}{6} = 1 - \frac{x+2}{7} \]

we find a common denominator, which is 42. Multiply every term by 42 to clear the fractions:

\[ 42 \cdot \frac{x+6}{6} = 42 \cdot \left(1 - \frac{x+2}{7}\right) \]

This simplifies to:

\[ 7(x+6) = 42 - 6(x+2) \]

Expand both sides of the equation:

\[ 7x + 42 = 42 - 6x - 12 \]

Simplify the right side:

\[ 7x + 42 = 30 - 6x \]

Add \( 6x \) to both sides to get all terms involving \( x \) on one side:

\[ 7x + 6x + 42 = 30 \]

Combine like terms:

\[ 13x + 42 = 30 \]

Subtract 42 from both sides:

\[ 13x = 30 - 42 \]

\[ 13x = -12 \]

Divide both sides by 13 to solve for \( x \):

\[ x = -\frac{12}{13} \]

Final Answer