SOLUTION: find the range of values which the line y=kx+3 meets the curve xy+20=5y

Question 712086: find the range of values which the line y=kx+3 meets the curve xy+20=5y
Answer by josgarithmetic(39617) (Show Source): You can put this solution on YOUR website!
You should be able to substitute y=kx+3 into xy+20=5y and solve for k. That should help to find . Knowledge of the general solution to a quadratic equation suggests k must not be zero.

Carry out the steps you find
or

From that you may want to be sure to know how the discriminant should be greater than or equal to zero. Solve . Again use solution to quadratic equation:

or But those values should be first checked as critical points. The k quadratic should be greater than OR EQUAL TO ZERO. There are three intervals of k to check the discriminant expression as .

Pushing ahead according to testing k in those intervals, the critical points in decimalized form would be close to k at 0.10389 and at 0.385. I picked to check values 0, 0.2, and 0.4. The results of the expression>=0 went like this:

at 0: , yes.
at 0.2: , no.
at 0.4: , yes.

Based on that, it seems k should be this:

This was based on expecting complex values of k to cause lack of meaningful solutions to the original set of equations.
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