Subtracted 82507

I think two numbers. Their ratio is 5:7. If 3 is subtracted from the first and 7 from the second, then the ratio is 6:7. What number was I thinking?

Correct answer:

a =  15
b =  21

Step-by-step explanation:

$$\begin{gathered} a:b=5:7 \\ (a-3):(b-7)=6:7 \\ 7a=5b \\ 7(a-3)=6\cdot (b-7) \\ 7\cdot a=5\cdot b \\ 7\cdot (a-3)=6\cdot (b-7) \\ 7a-5b=0 \\ 7a-6b=-21 \\ Row2-Row1\to Row2 \\ 7a-5b=0 \\ -b=-21 \\ b=\frac{-21}{-1}=21 \\ a=\frac{0+5b}{7}=\frac{0+5\cdot 21}{7}=15 \\ a=15 \\ b=21 \end{gathered}$$

$$\begin{gathered} b=21 \\ \text{ Verifying Solution: } \\ {r}_1=a:b=15:21=\frac{5}{7}\doteq 0.7143=5:7 \\ {r}_2=(a-3)/(b-7)=(15-3)/(21-7)=\frac{6}{7}\doteq 0.8571 \end{gathered}$$

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