Potter Ingram
11/13/2023 · Junior High School
Solve the equation \( \frac{9}{a}+18=0 \) Express \( \frac{1}{s}+\frac{1}{t}+\frac{1}{s t} \) as a single fraction.
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Tutor-Verified Answer
Step-by-step Solution
To solve the equation
\[
\frac{9}{a} + 18 = 0,
\]
we first isolate the term involving \( a \):
\[
\frac{9}{a} = -18.
\]
Next, we can cross-multiply to eliminate the fraction:
\[
9 = -18a.
\]
Now, we solve for \( a \) by dividing both sides by -18:
\[
a = \frac{9}{-18} = -\frac{1}{2}.
\]
Now that we have \( a = -\frac{1}{2} \), we can proceed to express \( \frac{1}{s} + \frac{1}{t} + \frac{1}{st} \) as a single fraction.
Let’s denote \( \frac{1}{s} = x \) and \( \frac{1}{t} = y \). Then we have:
\[
\frac{1}{s} + \frac{1}{t} + \frac{1}{st} = x + y + xy.
\]
To combine these terms into a single fraction, we can find a common denominator, which is \( st \):
\[
x + y + xy = \frac{t}{st} + \frac{s}{st} + \frac{1}{st} = \frac{t + s + 1}{st}.
\]
Thus, we can express \( \frac{1}{s} + \frac{1}{t} + \frac{1}{st} \) as:
\[
\frac{t + s + 1}{st}.
\]
In conclusion, the solution to the equation is
\[
a = -\frac{1}{2},
\]
and the expression \( \frac{1}{s} + \frac{1}{t} + \frac{1}{st} \) as a single fraction is
\[
\frac{t + s + 1}{st}.
\]
Quick Answer
The solution to the equation is \( a = -\frac{1}{2} \), and the expression \( \frac{1}{s} + \frac{1}{t} + \frac{1}{st} \) as a single fraction is \( \frac{t + s + 1}{st} \).
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