Lesson Digit problems - Find the number using systems of linear equations

Digit problems - Find the number using systems of linear equations

Problem 1

If the decimal form of number is  "ab",  then


10a + b = 6b,   according to the condition.


It implies  10a  = 5b,   or   b = 2a.


The sum of the digit  is 6:

a + b = 6 = a + 2a = 3a  ====>  a = 2;  b = 4.


The number is  "ab" = 24.

Problem 2

Let the decimal form of the number is "ab".


Then 

10a + b = 2 + 8*(a+b)      ("A two-digit number is 2 more than 8 times the sum of its digits")

10a + b = 2 + 8a + 8b,

2a = 2 + 7b.      (1)



Also,

a = 6 + b          (2)      ("tens digit is 6 more than unit digit")



Substitute (2) into (1). You will get

12 + 2b = 2 + 7b,

10 = 5b  ====>  b = 2.


Then a = b+6 = 2 + 6 = 8,   and you get the answer.

Problem 3

Let "ab" be the decimal presentation of the number, so "b" is the "units" digit 
and "a" is the tens" digit.


Then the value of the original number is (10a+b),
while the value of the reversed digit number is (10b+a).


From the problem,

    a + b = 11,                (1)

    10a + b = 10b + a + 45.    (2)


Simplify equation (2) step by step

    10a + b - 10b - a = 45

      9a - 9b = 45

      9(a-b) = 45

        a - b = 45.


Thus we have this system of equations

    a + b = 11,    (3)

    a - b =  5.    (4)


To solve the system, add equations (3) and (4)

    2a = 11 + 5 = 16,  a = 16/2 = 8.


Then from (3)  b = 11 -a = 11 - 8 = 3.


ANSWER.  The number is  83.


CHECK.  The sum of the digits is 8 + 3 = 11;

        the difference of the numbers is  83 - 38 = 45.   ! correct !

Problem 4

Let us present the unknown number N in decimal form via its "tens" digit "t"  and  "ones" digit "u"  N = 10*t + u.


Then the condition says

t = u + 5                   (1)
10*t + u = 3*t*u - 14.      (2)


It is your system of equations to solve.

From (1), substitute the expression for "t" into equation (2). You will get

10*(u+5) + u = 3*(u+5)*u - 14,

10u + 50 + u = 3u^2 + 15u - 14,

3u^2 + 4u - 64 = 0,

 =  = .


The only integer positive solution is u = 4.

Then   t = u+5 = 9.


The number is  94.