. the sum of the digits of a three-digits number is 16.the number formed when the hundred digit and the unit digit are interchanged is 99 less than the number.

Question

1. the sum of the digits of a three-digits number is 16.the number formed when the hundred digit and the unit digit are interchanged is 99 less than the number. The sum of the hundred digit and the unit digit is 2 less than the tens digit. Find the number.

 

2.solve each of the following system

i. {y-x=16

   {2x^2+y=27

 

ii. { y-x=10

    {xy+y^2=28

Answer 1

1. Let the number be xyz

The sum of the digits is 16 so:

 

x+y+z=16. equation 1

 

If we interchange the 100s digit and unit digit

the number is zyx

The value of zyx is 100z+10y+x

The value of xyz is 100x+10y+z

Since the number formed when we swap the

100s digit with the units digit is 99 less we have:

 

100z+10y+x = 100x+10y+z-99

Putting like terms together gives:

99x -99z = 99

99(x-z)=99

x-z=1     equation 2

 

The sum of the 100s digit and the units digit

is 2 less than the 10s digit so:

x+z=y-2

x-y+z=-2.    equation 3

 

We have 3 equations from information given:

 

x+y+z=16.    eqn 1

x-z=1.           eqn 2

x-y+z=-2.      eqn 3

 

from eqn 2:  z =x-1.    eqn 2a

 

Plug that into eqn 1 and eqn 3

 

x+y+x-1=16

2x+y =17.       eqn1a

 

x-y+x-1=-2

2x-y=-1.         eqn 3a

 

add eqn 1a and eqn 3a

 

2x+y=17

2x-y=-1

-------------

4x=16

x = 4

 

z=x-1=4-1=3

 

x+y+z=16

4+y+3=16

y+7=16

y=9

 

our number xyz is 493

******************************

2i).  y-x=16.      eqn 1

      2x²+y=27.   eqn 2

 

y=16+x.   from eqn 1

Substitute that into eqn 2

 

2x²+16+x=27

2x² +x-11=0

 

quadratic eqn:  x = [-b ±√(b²-4ac)]/2a

a=2, b=1, c=-11

 

x= -1/4 ± [√((1-4(2)(-11))]/4

x= -1/4 ±(√89)/4

 

x = (-1+√89)/4 or (-1-√89)/4

**************************

2ii).  y-x=10.      eqn 1

        xy+y²=28. eqn 2

 

x=y-10 from eqn 1

 

(y-10)y+y²=28

y²-10y +y²=28

2y²-10y-28=0

2(y²-5y-14)=0

y²-5y-14=0

(y-7)(y+2)=0

 

y=7  or   y=-2

 

if y=7

using eqn 1.. y-x=10

7-x=10

x=-3

solution:  (-3, 7)

 

If y=-2

-2-x=10

x=-12

 

solution:  (-12, -2)

 

There are two solutions to this system

which makes sense because we are dealing

with an equation of order 2.  Check by 

plugging both solutions into equation 1 and

equation 2 to verify they are both correct.

 

Good luck.  Hope this helps.