Let x be the first digit (tens place) of the original number and y the second digit (ones place) - such that the original number is xy.
The reverse of the original number, then, is yx (y is in the tens place, while x in the ones place).
[# note that xy and yx do not denote multiplication in this case, but simply the position of the values ]
The value of the first number then is 10x + y while the value of the second number is 10y + x.
We know that the original number is seven times the sum of its digits. This translates to `10x + y = 7(x + y).`
We also know that the number formed by reversing the original number is less than the original number by 18. This translates to: `10y + x = 10x + y - 18.`
The first equation simplifies to `3x-6y = 0.`
The second equation simplifies to `-x + y = -2.`
We now use the two equations to solve for x and y. Let us do both elimination and substitution.
Using the elimination method, we first multiply the second equation by 3 to get `-3x + 3y = -6.`
We then add the first equation to this new equation to get the value of y (x is cancelled since the first equation has 3x, while the second has -3x): `-3y = -6rArr y = 2.`
To get x, we simply substitute y to either equation. Let us use the second equation since it is simpler: `-x + 2 = -2 rArr -x = -4 rArr x = 4.`
Now, let us try to use the substitution method. First, let us isolate y from the first equation (you can also isolate x first, it does not really matter): `y = x/2.`
Then, we substitute this to the second equation: `-x + y = -2 rArr -x + x/2 = -2 rArr -2x + x = -4 rArr -x = -4 rArr x = 4.`
Now, we just solve for y by substituting the value of x we got to either the first or second equation. Again, let us use the second equation since it is simpler: `-x + y = -4 rArr -4 + y = -2 rArr y = 2.`
Note that we got the same answers for both methods.
The answer appears to be 42. Let us check. We know that the number is seven times the sum of its digits. The sum of the digits is 4+2 = 6. Then, 6*7 = 42. Hence, the condition is satisfied. The reverse of the number is 24. Then, 42 - 24 = 18. Which satisfies the second requirement.
The original number is 42.