Answer:
a = 7/3
Explanation:
You want the solution to the rational equation 6 = (3a+6)/a +1/a.
Graphical solution
It is convenient to rewrite the equation to the form f(a) = 0, so the x-intercepts provided by the graphing calculator are the solutions to the equation. Doing this, we have ...
f(a) = 6 -((3a +6)/a +1/a)
The attached graph shows the solution to be ...
a = 7/3 = 2 1/3
Algebraic solution
Multiplying the equation f(a) = 0 by 'a', we have ...
6a -((3a+6) +1) = 0
6a -3a -7 = 0 . . . . . . . . eliminate parentheses
3a = 7 . . . . . . . . . . add 7
a = 7/3 . . . . . . divide by 3
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Additional comment
The denominator cannot be zero, so any solution of a=0 is automatically excluded as being extraneous. Fortunately, there is no such solution here. Often, extraneous solutions can be avoided by the procedure used here. Effectively, we have written the equation as ...
If there were a factor of 'a' in the numerator, then considering the numerator alone would give a=0 as one of the solutions. However, that numerator factor would cancel the 'a' in the denominator, leaving an equation with no extraneous solutions.
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