Let [tex] a,b,c [/tex] be the three numbers. We know that the first number is 7 more than the third, which means [tex] a = c+7 [/tex]
Also, we know that the second number is 2 times the third, which means [tex] b = 2c [/tex]
Finally, we know that the three numbers sum up to 75, but we can rewrite the sum in terms of the third numbers, because the two previous equations gave us a way to express [tex] a [/tex] and [tex] b [/tex] in terms of [tex] c [/tex]:
[tex] a+b+c = 75 \iff (c+7)+(2c)+c=75 \iff 4c+7 = 75 \iff 4c = 68 \iff c = 17 [/tex]
Now, we can use those same equations to deduce the other two numbers:
[tex] a = c+7 \implies a = 17+7 = 24,\quad b = 2c \implies b = 2\cdot 17 = 34 [/tex]
