Does −6y+4(2y+7)=14 (8y+16)+24

have one solution, no solution, or infinitely many solutions?

1 answer

To determine whether the equation \(-6y + 4(2y + 7) = 14(8y + 16) + 24\) has one solution, no solution, or infinitely many solutions, we'll simplify both sides of the equation step by step.

  1. Expand both sides of the equation:

    Left side: \[ -6y + 4(2y + 7) = -6y + 8y + 28 = 2y + 28 \]

    Right side: \[ 14(8y + 16) + 24 = 112y + 224 + 24 = 112y + 248 \]

  2. Set the expanded expressions equal to each other: \[ 2y + 28 = 112y + 248 \]

  3. Rearranging to isolate \(y\): \[ 2y - 112y = 248 - 28 \] \[ -110y = 220 \]

  4. Solve for \(y\): \[ y = \frac{220}{-110} = -2 \]

Since we found a specific value for \(y\), the equation \(-6y + 4(2y + 7) = 14(8y + 16) + 24\) has exactly one solution: \(y = -2\).

Therefore, the answer is that the equation has one solution.