# Problem of the Week

## Updated at Jun 13, 2022 4:09 PM

To get more practice in equation, we brought you this problem of the week:

How can we solve the equation $$\frac{5}{2+y}+\frac{20}{y}=\frac{23}{3}$$?

Check out the solution below!

$\frac{5}{2+y}+\frac{20}{y}=\frac{23}{3}$

 1 Multiply both sides by the Least Common Denominator: $$3y(2+y)$$.$15y+60(2+y)=23y(2+y)$2 Simplify.$75y+120=46y+23{y}^{2}$3 Move all terms to one side.$75y+120-46y-23{y}^{2}=0$4 Simplify  $$75y+120-46y-23{y}^{2}$$  to  $$29y+120-23{y}^{2}$$.$29y+120-23{y}^{2}=0$5 Multiply both sides by $$-1$$.$23{y}^{2}-29y-120=0$6 Split the second term in $$23{y}^{2}-29y-120$$ into two terms.1 Multiply the coefficient of the first term by the constant term.$23\times -120=-2760$2 Ask: Which two numbers add up to $$-29$$ and multiply to $$-2760$$?$$40$$ and $$-69$$3 Split $$-29y$$ as the sum of $$40y$$ and $$-69y$$.$23{y}^{2}+40y-69y-120$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$23{y}^{2}+40y-69y-120=0$7 Factor out common terms in the first two terms, then in the last two terms.$y(23y+40)-3(23y+40)=0$8 Factor out the common term $$23y+40$$.$(23y+40)(y-3)=0$9 Solve for $$y$$.1 Ask: When will $$(23y+40)(y-3)$$ equal zero?When $$23y+40=0$$ or $$y-3=0$$2 Solve each of the 2 equations above.$y=-\frac{40}{23},3$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$y=-\frac{40}{23},3$DoneDecimal Form: -1.739130, 3y=-40/23,3