...from the first number the new number is 4 more than 5 times the sum of the digit in the first number. Find the number?

## Responses (2)

This is about a set of simultaneous equations.

1st digit := a

2nd digit := b

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1. The sum of a two digit number and a number formed by interchanging the digits is 110:

10a + b + 10b + a = 110 ;; simplify

11a + 11b = 110

1) a + b = 10. (First equation)

2. If 10 is subtracted from the first number, then the new number is 4 more than 5 times the sum of the digits in the first number:

10a + b - 10 = 5 (a + b) + 4 ;; simplify

10a + b - 10 = 5a + 5b + 4

2) 5a - 4b = 14. (Second equation)

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Solve the system of equations:

1) a + b = 10 ;; times 5

1a) 5a + 5b = 50

2) 5a - 4b = 14 ;; 1a) - 2) subtract eqn 2 from eqn 1

5a - 5a + 5b - (-4b) = 50 - 14

9b = 36

b = 4.

And, as a + b = 10 it follows

a = 6.

Check the results::

64 + 46 = 110

and

64 - 10 = 30 + 20 + 4

54 = 54.

That's it.