Solve the rational equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) [tex]$2+\frac{9}{r-3}=\frac{3 r}{r-3}$[/tex]
Answer (2)
Identify that r cannot be 3.
Multiply both sides of the equation by ( r − 3 ) .
Simplify the equation to 2 r + 3 = 3 r .
Solve for r to get r = 3 , which is an extraneous solution. Therefore, the equation has no solution. NOSO LU T I ON
Explanation
Analyze the equation We are given the rational equation 2 + r − 3 9 = r − 3 3 r and asked to solve for r . First, we need to identify any values of r that would make the denominator equal to zero, as these values would be excluded from the solution set. In this case, r − 3 = 0 when r = 3 , so r cannot be equal to 3.
Eliminate the fractions To solve the equation, we can multiply both sides by ( r − 3 ) to eliminate the fractions. This gives us: ( r − 3 ) ( 2 + r − 3 9 ) = ( r − 3 ) ( r − 3 3 r ) 2 ( r − 3 ) + 9 = 3 r
Expand and simplify Now, we expand and simplify the equation: 2 r − 6 + 9 = 3 r 2 r + 3 = 3 r
Isolate r Next, we subtract 2 r from both sides of the equation: 2 r + 3 − 2 r = 3 r − 2 r 3 = r
Check for extraneous solutions We found that r = 3 . However, we noted earlier that r cannot be equal to 3 because it would make the denominators in the original equation equal to zero. Therefore, r = 3 is not a valid solution.
Conclusion Since r = 3 is not a valid solution, there is no solution to the given rational equation.
Examples
Rational equations are used in many real-world applications, such as calculating the time it takes to complete a task when working together. For example, if one person can complete a job in x hours and another person can complete the same job in y hours, then the equation x 1 + y 1 = t 1 can be used to find the time t it takes for them to complete the job together. Solving such equations helps in optimizing processes and resource allocation.
The solution to the rational equation 2 + r − 3 9 = r − 3 3 r leads to the extraneous solution r = 3 , which is not valid since it makes the denominator zero. Therefore, there is no solution to this equation. The final answer is NO SO LU T I ON .
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