Getting rid of numerator in a fraction equation

Question

Im doing Okun's law for economics.

Which is $\dfrac{Y^*-357665}{Y^*} = 1.5(5.1-4.75)$

how do I find $Y^*$?

Answer 1

First of all, $1.5(5.1 - 4.75) = 0.525$ so we have

$$\frac{Y^* - 357665}{Y^*} = 0.525.$$

Now the goal is to obtain $Y^*$ but the fraction is complicating things, so we should try to get rid of it. As the opposite of division is multiplication, we should multiply by $Y^*$, but to maintain the equality between the two sides, we have to multiply both sides by $Y^*$. That is,

$$\frac{Y^* - 357665}{Y^*}\times Y^* = 0.525\times Y^*$$

which simplifies to

$$Y^* - 357665 = 0.525Y^*.$$

Can you find $Y^*$ from here?

Added Later: Evidently not, but no matter, let's press on.

Now we have two $Y^*$ terms and a constant. As we are trying to find $Y^*$, let's put all the $Y^*$ terms on one side (let's say the left). We do this by subtracting $0.525Y^*$ from both sides.

$$Y^* - 357665 - 0.525Y^* = 0.525Y^* - 0.525Y^*.$$

Combining the $Y^*$ terms, this becomes

$$0.475Y^* - 357665 = 0.$$

Can you find $Y^*$ from here?

Added Even Later: Given the final equation you can find $Y^*$ (well done), but it seems that the combination of $Y^*$ terms wasn't clear so let me explain that a bit more.

We have the equation

$$Y^* - 357665 - 0.525Y^* = 0.525Y^* - 0.525Y^*$$

from above. Now, on the right hand side, we are subtracting $0.525Y^*$ from itself, so we are left with zero (let me know if I need to explain this), so we have

$$Y^* - 357665 - 0.525Y^* = 0.$$

The question is now, how does the left hand side become $0.475Y^* - 357665$? As it doesn't matter whether I subtract $357665$ first and then subtract $0.525Y^*$, or vice versa, I can rewrite this equation as

$$Y^* - 0.525Y^* - 357665 = 0$$

which is beneficial as now the two $Y^*$ terms are adjacent. I want to combine the first two terms. I do this by taking out a common factor of $Y^*$ (let me know if I need to explain this) as follows

$$(1 - 0.525)Y^* - 357665 = 0$$

where I have used the fact that $Y^* = 1Y^*$. Now $1 - 0.525 = 0.475$ so the equation becomes

$$0.475Y^* - 357665 = 0$$

as above.