Final answer:
To find the three numbers that satisfy the given conditions, a system of equations was set up and solved. The correct values for the first, second, and third numbers are found to be 2, 5, and 3, which corresponds to option A.
Step-by-step explanation:
To find the three numbers based on the given conditions, we can set up a system of equations based on the provided statements. Let's define the first number as x, the second number as y, and the third number as z.
The sum of the three numbers is 10:
1. x + y + z = 10
The sum of twice the first number, 5 times the second number, and 6 times the third number is 51:
2. 2x + 5y + 6z = 51
The difference between 8 times the first number and the second number is 3:
3. 8x - y = 3
By solving these three equations simultaneously, we can find the values of x, y, and z which represent the first, second, and third numbers respectively.
By rearranging equation 3, we get: y = 8x - 3.
Let's plug the value of y into equation 1:
x + 8x - 3 + z = 10 => 9x + z = 13 => z = 13 - 9x
Now we substitute the values of y and z into equation 2:
2x + 5(8x - 3) + 6(13 - 9x) = 51
After simplifying and solving for x, we find that:
x = 2,
This allows us to then find the values for y and z:
y = 5,
z = 3
Option A is the correct answer: First number: 2, Second number: 5, Third number: 3.