The original equation was without the added $\frac{2x-1}{2x-1}$. I don't understand how it simplified the equation. How did we get from step 2 to step 3, where all fractions within fractions vanished?
2026-03-29 23:26:04.1774826764
How did multiplying the numerator and denominator by the same fraction really simplified this equation?
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They just distributed the $2x-1$ on the top and did the same to the bottom. This was helpful because we had fractions within fractions and in this particular case, eliminating them simplifies things a lot.
In the numerator: $$(2x-1)\cdot \left(\dfrac{4+5x}{2x-1}+4\right)=(2x-1)\cdot\left(\dfrac{4+5x}{2x-1}\right)+(2x-1)\cdot 4$$ $$=4+5x+8x-4=13x$$ In the denominator: $$(2x-1)\cdot\left(2\left(\dfrac{4+5x}{2x-1}\right)-5\right)=(2x-1)\cdot 2\left(\dfrac{4+5x}{2x-1}\right)-(2x-1)\cdot 5$$ $$=2(4+5x)-(10x-5)=8+10x-10x+5=13$$ and the other simplifications follow.