%2F(x%5E(2)-3x)-(6)%2F(x-3)%3D(5)%2F(x).jpeg "(18)/(x^(2)-3x)-(6)/(x-3)=(5)/(x)")
Solving the Rational Equation: (18)/(x^(2)-3x)-(6)/(x-3)=(5)/(x)
This article will guide you through the steps of solving the rational equation: (18)/(x^(2)-3x)-(6)/(x-3)=(5)/(x).
Step 1: Factor the Denominators
- The first denominator, x^(2)-3x, can be factored as x(x-3).
Now the equation becomes: (18)/(x(x-3))-(6)/(x-3)=(5)/(x)
Step 2: Find the Least Common Multiple (LCM)
- The LCM of the denominators x(x-3), x-3, and x is x(x-3).
Step 3: Multiply Each Term by the LCM
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Multiply both sides of the equation by x(x-3):
- (18)/(x(x-3)) * x(x-3) - (6)/(x-3) * x(x-3) = (5)/(x) * x(x-3)
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This simplifies to:
- 18 - 6x = 5(x-3)
Step 4: Simplify and Solve for x
- Expand the right side: 18 - 6x = 5x - 15
- Combine like terms: 33 = 11x
- Solve for x: x = 3
Step 5: Check for Extraneous Solutions
- We need to check if x = 3 is a valid solution by plugging it back into the original equation.
- Notice that if x = 3, the denominators of the first two terms become zero, which is undefined.
- Therefore, x = 3 is an extraneous solution, and the equation has no solution.
Conclusion
The rational equation (18)/(x^(2)-3x)-(6)/(x-3)=(5)/(x) has no solution. This is because the potential solution we found, x = 3, is an extraneous solution that makes the denominator of the original equation equal to zero.