Answered

What is the solution to the equation $\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}$ ?


A$m=3$
B$m=6$
Call real numbers
Dno solution

Verified Answer :

  • Analyze the equation and determine the domain constraints \(m eq 3, -3\).
  • Simplify the equation by finding a common denominator: \(\frac{3}{m+3} + \frac{m}{m-3} = \frac{m^2+9}{(m+3)(m-3)}\).
  • Clear the fractions and solve the resulting linear equation to find \(m=3\).
  • Recognize that \(m=3\) is an extraneous solution due to the initial domain constraints, concluding there is \(\boxed{\text{no solution}}\).

Explanation

  1. Analyzing the Equation and Domain Constraints

We are given the rational equation:
\[\frac{3}{m+3} - \frac{m}{3-m} = \frac{m^2+9}{m^2-9}.\]
First, identify the domain of the expression. The denominators must not be zero:

  1. \(m+3 eq 0 \implies m eq -3\)
  2. \(3-m eq 0 \implies m eq 3\)
  3. \(m^2-9 eq 0 \implies (m-3)(m+3) eq 0 \implies m eq 3, -3\)

Thus, the domain is all real numbers except \(m=3\) and \(m=-3\).

  1. Simplifying the Equation

To solve the equation, we rewrite the terms to have a common denominator. Observe that:
\[m^2-9 = (m-3)(m+3) = -(3-m)(m+3).\]
We can rewrite the second term by multiplying its numerator and denominator by \(-1\):
\[-\frac{m}{3-m} = - \frac{m(-1)}{(3-m)(-1)} = - \frac{-m}{m-3} = \frac{m}{m-3}.\]
Substituting this back into our original equation gives:
\[\frac{3}{m+3} + \frac{m}{m-3} = \frac{m^2+9}{(m+3)(m-3)}.\]

  1. Solving for m

Now, multiply the entire equation by the common denominator \((m+3)(m-3)\) to clear the fractions:
\[(m+3)(m-3) \left( \frac{3}{m+3} + \frac{m}{m-3} \right) = (m+3)(m-3) \left( \frac{m^2+9}{(m+3)(m-3)} \right).\]
Expanding the left side:
\[3(m-3) + m(m+3) = m^2+9.\]
\[3m - 9 + m^2 + 3m = m^2+9.\]
Simplifying the left side:
\[m^2 + 6m - 9 = m^2 + 9.\]
Subtracting \(m^2\) from both sides leads to:
\[6m - 9 = 9.\]
\[6m = 18.\]
\[m = 3.\]

  1. Conclusion and Verification

We found the potential solution \(m=3\). However, recall our domain constraint from the first step: the expressions \(\frac{m}{3-m}\) and \(\frac{m^2+9}{m^2-9}\) are undefined when \(m=3\), as they lead to a zero denominator. Since the only candidate value \(m=3\) is excluded from the domain, there is no valid solution to the equation. Therefore, the correct option is "no solution".

Examples

In engineering and physics, rational equations often arise when calculating equivalent resistance in parallel circuits or modeling flow rates. However, it is vital to always verify if the calculated values fall within the physical constraints of the system. Just as in this problem, a mathematical solution might appear valid, but if it requires dividing by zero (which in this context might represent a physical impossibility like a vacuum or an open circuit), it must be rejected as an extraneous solution.

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