Question

Solve the equation by multiplying each side by the least common denominator. $$\frac{x}{x+9}=\frac{9}{x+9}+2$$

Short answer

The solution to the equation is \(x = -27\).

Step-by-step solution

Multiply by the LCD

Multiply each side of the equation by the LCD \(x + 9\). This gives:\[(x+9) \cdot \frac{x}{x+9} = (x+9) \cdot \left (\frac{9}{x+9} + 2 \right)\]

Simplify the equation

Simplify the equation by cancelling out the denominator on the left side and distributing on the right side. This gives:\[x = 9 + 2(x+9)\]

Distribute on the right side

Distribute the '2' to both the 'x' and '9' on the right side of the equation, which gives:\[x = 9 + 2x + 18\]

Combine like terms and isolate the variable

Combine like terms on the right side and then isolate 'x' by subtracting '2x' from both sides, which gives:\[-x = 27\]

Solve the equation

Finally, we divide both sides by -1 to solve for 'x', which gives:\[x = -27\]