To find the original number, let's first represent it as a two-digit number with the tens digit as "x" and the units digit as "y". Therefore, the original number can be written as 10x + y.
According to the given information:
1) "The value of the number is eight times the sum of its digits:"
So, the equation is: 10x + y = 8(x + y)
2) "If the digits of the number are reversed, the result is 45 less than the original number:"
When the digits are reversed, the new number becomes 10y + x. Therefore, we can write the equation as: 10x + y - 45 = 10y + x
Now, let's solve these two equations simultaneously to find the values of x and y:
Equation 1: 10x + y = 8(x + y)
Expanding: 10x + y = 8x + 8y
Simplifying: 2x - 7y = 0
Equation 2: 10x + y - 45 = 10y + x
Expanding: 9x - 9y = 45
Simplifying: x - y = 5
Now, we have a system of equations:
2x - 7y = 0 ----(A)
x - y = 5 ----(B)
To solve, we can multiply equation (B) by 2 to eliminate the x variable:
2(x - y) = 2(5)
2x - 2y = 10
Now, we can substitute this equation into equation (A):
2x - 7y = 0
(2x - 2y) - 5y = 0
10 - 5y = 0
-5y = -10
y = 2
By substituting the value of y into equation (B):
x - 2 = 5
x = 7
Therefore, the original two-digit number is 10x + y = 10(7) + 2 = 72.