Question 919435
let x = the tens digit
let y = the units digit


10x + y = 3 * (x + y)
simplify to get:
10x + y = 3x + 3y
subtract 3x from both sides of the equation to get:
7x + y = 3y
subtract y from both sides of the equation to get:
7x = 2y
divide both sides of the equaation by 2 to get:
y = 7x/2


10x + y = 10y + x - 45
subtract x from both sides of the equation to get:
9x + y = 10y - 45
subtract 10y from both sides of the equation to get:
9x - 9y = -45
replace y with 7x/2 to get:
9x - 9 * (7x/2) = -45
simplify to get:
9x - 63x / 2 = -45
multiply both sides of the equation by 2 to get:
18x - 63x = -90
combine like terms to get:
-45x = -90
divide both sides of the equation by -2 to get:
x = 2


y = 7x/2
replace x with 2 to get:
y = (7*2)/2
simplify to get:
y = 7


that's your solution.
x = 2
y = 7
10x + y = 27


10x + y = 3 * (x+y) becomes
10*2 + 7 = 3 * (2 + 7) which becomes:
20 + 7 - 3 * 9 which becomes:
27 = 27 which is true.


10x + y = 10y + x - 45 becomes:
10 * 2 + 7 = 10 * 7 + 2 - 45 which becomes:
20 + 7 = 70 + 2 - 45 which becomes:
27 = 72 - 45 which becomes:
27 = 27 which is true.


everything checks out so the solution is good.


you could also have done a simple logic test up front which would have told you that y had to be equal 7 and x have to be equal to 2.


this is because y = 7x/2

when x = 2, y equal 7.
x had to be between 0 and 9
y had to be between 0 and 9.
x and y both had to be integers.
the only value of x that gave a valid value of y was x = 2.
all others were either out of range of not integers.