\frac{2}{2-y}+\frac{3}{y+2}=\frac{2 y}{y^2-4}
Answer (2)
Multiply both sides of the equation by ( y − 2 ) ( y + 2 ) to eliminate the denominators.
Simplify the equation to − 2 ( y + 2 ) + 3 ( y − 2 ) = 2 y .
Combine like terms to get y − 10 = 2 y .
Solve for y to get y = − 10 . The final answer is − 10 .
Explanation
Understanding the Problem We are given the equation 2 − y 2 + y + 2 3 = y 2 − 4 2 y We want to solve for y . First, notice that y 2 − 4 = ( y − 2 ) ( y + 2 ) . Thus, we can rewrite the equation as 2 − y 2 + y + 2 3 = ( y − 2 ) ( y + 2 ) 2 y Note that y = 2 and y = − 2 , otherwise the denominators would be zero, making the fractions undefined.
Eliminating the Denominators To eliminate the denominators, we multiply both sides of the equation by ( y − 2 ) ( y + 2 ) . Note that 2 − y = − ( y − 2 ) , so 2 − y 1 = − y − 2 1 .
Multiplying both sides by ( y − 2 ) ( y + 2 ) gives 2 − y 2 ( y − 2 ) ( y + 2 ) + y + 2 3 ( y − 2 ) ( y + 2 ) = ( y − 2 ) ( y + 2 ) 2 y ( y − 2 ) ( y + 2 ) 2 ( 2 − y y − 2 ) ( y + 2 ) + 3 ( y − 2 ) = 2 y Since 2 − y y − 2 = − 1 , we have − 2 ( y + 2 ) + 3 ( y − 2 ) = 2 y − 2 y − 4 + 3 y − 6 = 2 y y − 10 = 2 y
Solving for y Now, we solve for y :
y − 10 = 2 y Subtract y from both sides: − 10 = y So, y = − 10 .
Checking the Solution We need to check if the solution y = − 10 is valid. We must have y = 2 and y = − 2 . Since − 10 is not equal to 2 or -2, the solution is valid. Therefore, the solution is y = − 10 .
Examples
When solving equations involving rational expressions, it's crucial to identify values that make the denominator zero, as these values are excluded from the solution set. This concept is widely used in engineering when designing systems where certain parameters cannot reach specific values to avoid instability or failure. For example, in electrical circuit design, we must ensure that the impedance of a circuit never reaches zero to prevent a short circuit. Similarly, in mechanical engineering, resonance frequencies must be avoided to prevent structural damage.
The solution to the equation 2 − y 2 + y + 2 3 = y 2 − 4 2 y is y = − 10 . This value does not make any denominators zero, so it is a valid solution. Thus, the final answer is − 10 .
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