Question

What is the solution to the equation StartFraction y Over y minus 4 EndFraction minus StartFraction 4 Over y + 4 EndFraction = StartFraction 32 Over y squared minus 16 EndFraction?
y = –4 and y = 4
y = 0
all real numbers
no solution

Answer 1

\[\frac{ y }{ y-4 }-\frac{ 4 }{ y+4 }=\frac{ 32 }{ y^{2}-16 }\]

y^2 - 16 is a difference of squares. so it can be factored into (y+4)(x-4)

\[\frac{ y }{ y-4 }-\frac{ 4 }{ y+4 }=\frac{ 32 }{ (y+4)(y-4)}\]
now, notice how the third denominator has the two factors (y+4) and (y-4). each denominator on the left side has one of these factors. for each left fraction, multiply by the appropriate factor to convert the denominator to (y+4)(y-4). so for y / (y-4), it's missing a y + 4 in the denominator, so we multiply the whole fraction by (y+4)/(y+4).

\[\frac{ y(y+4) }{ (y+4)(y-4) }-\frac{ 4 }{ y+4 }=\frac{ 32 }{ (y+4)(y-4) }\]
same logic with the second fraction. multiply by (y-4)/(y-4)

\[\frac{ y(y+4) }{ (y+4)(y-4) }-\frac{ 4(y-4) }{ (y+4)(y-4) }=\frac{ 32 }{ (y+4)(y-4) }\]

now that all the denominators are equal, you can eliminate the denominators by multiplying both sides by (y+4)(y-4)

y(y+4) - 4(y-4) = 32

finish by distributing, combining like terms, and isolating y