The sum of all three numbers is 12. the sum of twice the first number, 4 times the second number, and 5 times the third number is 35. the difference between 8 times the first number and the second number is 52. what are the three numbers?

The numbers are 7, 4 and 1

Step-by-step explanation:

Let the numbers =x, y, z.

The sum of all three numbers is 12, x+y+z=12i

The sum of twice the first number, 4 times the second number, and 5 times the third number is 35, 2x+4y+5z=35ii

The difference between 8 times the first number and the second number is 52, 8x-y=52iii

Now, from (i),z =1-x-yiv

substitute (iv) in (ii),

2x+4y+5(12-x-y)=35

2x+4y+60-5x-5y=35

collect like terms

-3x-y=-25

that is, 3x-y=25v

solving (iii) and (v) simultaneously (add iii and v), we have

11x=77

divide both sides by 11 to obtain x=7

Put x=7 in (v)

3(7)+y=25

21+y=25

implies y=25-21=4

put x=7 and y=4 in (iv)

z=12-7-4=1

Hence, the numbers are 7, 4 and 1.

may i see the picture? it would a lot, xox