Value of an expression

[PGGD]

Hello! Please help me, I can't solve this task:
"Find such values of 'y' that the value of expression (yx+3)/4x-2 does not depend on the value of 'x'."
Thanks.

 

[Deleted member 4993]

Hello! Please help me, I can't solve this task:
"Find such values of 'y' that the value of expression (yx+3)/4x-2 does not depend on the value of 'x'."
Thanks.

is your function

\(\displaystyle \dfrac{yx + 3}{4x} \ - \ 2\)

or

\(\displaystyle \dfrac{yx + 3}{4x \ - \ 2}\)

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[HallsofIvy]

Assuming this is \(\displaystyle \frac{yx+ 3}{4x- 2}\), since both numerator and denominator are of first degree, "long division" will give a number plus a fraction with denominator 4x-2. In order that this "not depend on x", the numerator of that fraction (the "remainder" when yx+ 3 is divided by 4x- 2) must be 0.

You could also do this by noting that 4x- 2= 4(x- 1/2) so that \(\displaystyle \frac{yx+ 3}{4x- 2}= \frac{1}{4}\frac{yx+ 3}{x- 1/2}\) and the remainder when polynomial, p(x) is divided by x- 1/2 is p(1/2).