What is the solution to the equation \( \frac{2}{3} x+1=\frac{1}{6} x-7 ? \) \( x=-16 \) \( x=-4 \) \( x=4 \) \( x=16 \) \( x \)

To solve the equation \( \frac{2}{3} x + 1 = \frac{1}{6} x - 7 \), we first want to eliminate the fractions by finding a common denominator. The least common multiple of 3 and 6 is 6. Multiplying every term by 6 gives us: \[ 6 \left( \frac{2}{3}x \right) + 6(1) = 6 \left( \frac{1}{6}x \right) - 6(7) \] This simplifies to: \[ 4x + 6 = x - 42 \] Now, let's solve for \( x \) by isolating it. Rearranging gives us: \[ 4x - x = -42 - 6 \] This leads us to: \[ 3x = -48 \] Finally, dividing both sides by 3: \[ x = -16 \] So, the correct solution is \( x = -16 \). For a bit of extra fun, did you know that solving linear equations is like a balancing act? You have to keep the equation "balanced" by doing the same thing to both sides. It’s like a seesaw – if one side goes up, you need to adjust the other side to keep it level! And if you’re looking to sharpen your problem-solving skills, remember to check your work! Substitute your solution back into the original equation. It’s a great habit that helps catch any little errors and reinforces your confidence in your answer. You’ll feel like a math superhero!