Solve the following equation:
[tex]\frac{3}{2 b-5}-\frac{4}{b-3}=0[/tex]
Answer (2)
Add b − 3 4 to both sides: 2 b − 5 3 = b − 3 4 .
Cross-multiply: 3 ( b − 3 ) = 4 ( 2 b − 5 ) .
Expand: 3 b − 9 = 8 b − 20 .
Solve for b : b = 5 11 .
5 11
Explanation
Problem Analysis We are given the equation 2 b − 5 3 − b − 3 4 = 0 and we want to solve for b .
Isolating Terms First, we add b − 3 4 to both sides of the equation to isolate the terms with b :
2 b − 5 3 = b − 3 4
Cross-Multiplication Next, we cross-multiply to eliminate the fractions: 3 ( b − 3 ) = 4 ( 2 b − 5 )
Expanding the Equation Now, we expand both sides of the equation: 3 b − 9 = 8 b − 20
Isolating b We want to isolate b , so we subtract 3 b from both sides: − 9 = 5 b − 20
Further Isolation Next, we add 20 to both sides: 11 = 5 b
Solving for b Finally, we divide both sides by 5 to solve for b :
b = 5 11
Checking the Solution We should check if this solution is valid by plugging it back into the original equation. We need to make sure that the denominators are not zero. 2 b − 5 = 2 ( 5 11 ) − 5 = 5 22 − 5 25 = − 5 3 = 0 b − 3 = 5 11 − 3 = 5 11 − 5 15 = − 5 4 = 0 Since the denominators are not zero, the solution is valid.
Final Answer Therefore, the solution to the equation is b = 5 11 .
Examples
When you're mixing a cleaning solution, the concentration of ingredients is crucial. This equation helps determine the exact amount of a substance needed to achieve the desired concentration, ensuring the solution works effectively without being too weak or too strong. By solving for the variable, you can fine-tune the recipe for optimal results.
To solve the equation 2 b − 5 3 − b − 3 4 = 0 , we follow steps to isolate b and ultimately find that b = 5 11 . We verified that our solution does not lead to any division by zero. Thus, the final answer is 5 11 .
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