What is the solution to the equation $\frac{1}{h-5}+\frac{2}{h+5}=\frac{16}{h^2-25}$?
A. $h=\frac{11}{3}$
B. $h=5$
C. $h=7$
D. $h=\frac{21}{2}$
Answer (2)
Factor the denominator on the right side of the equation.
Multiply both sides by ( h − 5 ) ( h + 5 ) to clear the fractions.
Simplify and solve the resulting linear equation.
The solution to the equation is 7 .
Explanation
Problem Analysis We are given the equation h − 5 1 + h + 5 2 = h 2 − 25 16 and asked to find the solution for h .
Factoring and Restrictions First, notice that h 2 − 25 can be factored as ( h − 5 ) ( h + 5 ) . This means we can rewrite the equation as h − 5 1 + h + 5 2 = ( h − 5 ) ( h + 5 ) 16 . Also, we must have h = 5 and h = − 5 , otherwise the denominators would be zero, making the fractions undefined.
Eliminating Denominators To solve for h , we can multiply both sides of the equation by ( h − 5 ) ( h + 5 ) to eliminate the denominators: ( h − 5 ) ( h + 5 ) ( h − 5 1 + h + 5 2 ) = ( h − 5 ) ( h + 5 ) ( ( h − 5 ) ( h + 5 ) 16 ) . This simplifies to ( h + 5 ) + 2 ( h − 5 ) = 16.
Simplifying the Equation Now, we expand and simplify the left side of the equation: h + 5 + 2 h − 10 = 16. Combining like terms, we get 3 h − 5 = 16.
Isolating the Variable Next, we add 5 to both sides of the equation: 3 h = 16 + 5 , which gives 3 h = 21.
Solving for h Finally, we divide both sides by 3 to solve for h : h = 3 21 = 7. Since h = 7 is not equal to 5 or -5, it is a valid solution.
Final Answer Therefore, the solution to the equation is h = 7 .
Examples
Understanding how to solve rational equations is crucial in many fields, such as physics and engineering, where you might need to analyze circuits or fluid dynamics. For instance, when dealing with electrical circuits, you might encounter equations involving resistances in parallel, which can be expressed as rational functions. Solving these equations helps determine the overall resistance in the circuit, which is essential for designing and troubleshooting electrical systems. Similarly, in fluid dynamics, rational equations can arise when analyzing flow rates and pressures in pipes, allowing engineers to optimize the design of pipelines and hydraulic systems. The ability to manipulate and solve these equations ensures efficient and safe operation of various technological applications.
The solution to the equation is h = 7 . This was found by clearing the fractions and simplifying the resulting equation. The final answer confirms that 7 is valid since it's not equal to the values that would make the original denominators zero.
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