Solve: $\frac{m}{m+9}=\frac{63}{22}$.
Enter your answer as a fraction.
The given equation is \(\frac{m}{m+9} = \frac{63}{22}\). To solve for \(m\), we first identify the restriction on the variable. Since the denominator cannot be zero, we must have \(m+9 eq 0\), which means \(m eq -9\).
We can solve the rational equation by using cross-multiplication, which is equivalent to multiplying both sides by the least common multiple of the denominators, \(22(m+9)\). This gives:
\[ 22 \times m = 63 \times (m+9) \]
Now, expand the right side:
\[ 22m = 63m + 567 \]
To isolate \(m\), subtract \(63m\) from both sides of the equation:
\[ 22m - 63m = 567 \]
Combine like terms on the left side:
\[ -41m = 567 \]
Finally, divide both sides by \(-41\) to find the value of \(m\):
\[ m = -\frac{567}{41} \]
Since 567 is \(3^4 \times 7\) and 41 is a prime number, this fraction cannot be simplified further. Also, \(-\frac{567}{41} \approx -13.829\), which satisfies the condition \(m
eq -9\).
Rational equations like this one appear in various fields such as chemistry and physics. For example, if you are mixing two solutions to achieve a specific concentration, you might set up an equation where the ratio of components represents the final concentration. Solving for an unknown volume or concentration allows you to determine exactly how much of a substance is needed to meet a target ratio, ensuring accuracy in experiments or industrial processes.