Grades 11-12 Video Solutions 2010-11&12 Video Solutions 2010 problem29

Video Transcription

Question number 29. Function f, defined for the set of all positive real numbers and yielding real values, satisfies the equation 2 times f plus 3 times f evaluated at 2010 divided by x is equal to 5x for every positive value of x.  The value of f of 6 is then equal to which of the following? So let's see what happens when we let x be equal to 6. And then we have here the equation 2f evaluated at 6.  And that's equal to 5x but we still have 3 times f of 2010 divided by 6 and then that's equal to 5 times 6. And 2010 can be factored as 6 times 335.  So then we have an equation here that looks like 2f evaluated at 6 plus 3f evaluated at 335 is equal to 30. And this here looks like the next value of x that we should use.  So let now x be equal to 335 and we have then the equation 2f evaluated at 335 plus 3 times f. Here we have 2010 divided by 335 and that is equal to 5 times x which is 335.  So by our factorization what we're looking at is 2f of 335 plus 3 times f of 6 and that's equal to 5 times 335 which is 1675.  So we have then two equations here and two unknowns and we can solve this system of equations. And we can use substitution.  So from the first equation I will solve for one of the variables and then substitute that into the second equation.  So we have for example 3f of 335 is equal to 30 minus 2 times f of 6 like that. So f of 335 is equal to 30 minus 2 f of 6 all divided by 3.  So then we substitute this into the second equation. So we have in the second equation 2 times f of 335 is then added to 3 times f of 6 and that's 1675.  And we can substitute and solve for f of 6. So we have 2 thirds then 30 minus 2 f of 6 plus 3 f of 6 is 1675. And then we can keep working with this.  So 2 thirds of 30 is 10 minus 4 thirds f of 6 plus 3 times f of 6 and that's 1675.  So on the left hand side we have 3 over 9 thirds so subtracting I have 5 thirds f of 6 and then 1675 minus 10 is 1665 like that.  Which is obviously divisible by 5 and then we can just say that f of 6 is 3 fifths times 1665 and after computing here we obtain 993 as the answer.  So that is the value of f of 6 and we can now answer the question and the answer is A, 993.

Video Summary

Summary Not Available