SOLUTION: Need help with set up please!! I know the answer is 83 through playing with numbers, but I need to show it algebraically. thanks in advance The tens digit of a two-digit number

Question 724360: Need help with set up please!! I know the answer is 83 through playing with numbers, but I need to show it algebraically. thanks in advance
The tens digit of a two-digit number is 5 more than the units digit. If "the number" is divided by the sum of its digits, the partial quotient is 7 and the remainder is 6. Find "the number."
Found 3 solutions by KMST, ankor@dixie-net.com, josmiceli:
Answer by KMST(5399)   (Show Source): You can put this solution on YOUR website!
= tens digit
= units digit
so the value of the number is and
the sum of the digits is
The tens digit is 5 more than the units digit translates as

Now comes the hard part:
If "the number" is divided by the sum of its digits, the partial quotient is 7 and the remainder is 6 translates as
(dividend = divisor x quotient + remainder)

Another way to see that:
The number divided by is 7 plus the fraction that we get when we divide the remainder by

Multiplying times both sides of the equal sign we get

Anyway,
--> --> --> --> -->
Now we have a system of linear equations:
--> --> --> -->
--> -->
Answer by ankor@dixie-net.com(22740)   (Show Source): You can put this solution on YOUR website!
let a = the 10's digit
let b = the units
then
10a+b = "the number"
:
Write an equation for each statement
:
"The tens digit of a two-digit number is 5 more than the units digit."
a = b + 5
:
"If "the number" is divided by the sum of its digits, the partial quotient is 7 and the remainder is 6."
subtract the remainder (6) from the number to get an even 7
= 7
multiply both sides by (a+b)
10a + b - 6 = 7(a+b)
10a + b - 6 = 7a + 7b
10a - 7a = 7b - b + 6
3a = 6b + 6
simplify, divide by 3
a = 2b + 2
Replace a with (b+5)
b + 5 = 2b + 2
5 - 2 = 2b - b
b = 3,
then
a = 3 + 5
a = 8
:
83 is the number
:
:
Check this, divide 83 by 11, 7 a remainder of 6

Answer by josmiceli(19441)   (Show Source): You can put this solution on YOUR website!
Let = the units digit
Let = the tens digit
given:
(1)
(2)
------------------
Substitute (1) into (2)
(2)
(2)
Multiply both sides by
(2)
(2)
(2)
(2)
and, since
(1)
(1)
(1)
The number is 83
check answer:
(2)
(2)
(2)
(2)
(2)
OK

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