Solve for \( y \). \[ 4+\frac{1}{y-2}=-\frac{7}{y-5} \] If there is more than one solution, separate them with commas. ff there is no solution, click on "No solution". \( y=\square \)
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To solve the equation \[ 4+\frac{1}{y-2}=-\frac{7}{y-5}, \] first, let's eliminate the fractions by finding a common denominator. The common denominator is \((y-2)(y-5)\). Multiply both sides by this expression: \[ 4(y-2)(y-5) + (y-5) = -7(y-2). \] Now, let's distribute: \[ 4(y^2 - 7y + 10) + (y - 5) = -7y + 14. \] This simplifies to: \[ 4y^2 - 28y + 40 + y - 5 = -7y + 14. \] Combine like terms: \[ 4y^2 - 27y + 35 = -7y + 14. \] Now, add \(7y\) and subtract \(14\) from both sides: \[ 4y^2 - 27y + 7y + 35 - 14 = 0, \] which simplifies to: \[ 4y^2 - 20y + 21 = 0. \] To solve this quadratic equation, we can use the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -20\), and \(c = 21\). Calculating the discriminant: \[ b^2 - 4ac = (-20)^2 - 4 \cdot 4 \cdot 21 = 400 - 336 = 64. \] Now apply the quadratic formula: \[ y = \frac{-(-20) \pm \sqrt{64}}{2 \cdot 4} = \frac{20 \pm 8}{8}. \] This gives us two solutions: 1. \(y = \frac{28}{8} = 3.5\) 2. \(y = \frac{12}{8} = 1.5\) So, the solutions are: \( y = 3.5, 1.5 \)
