Question

The sum of three numbers is 6.If we multiply the third number by 2 and add the first number to the result,we get 7.By adding second and third numbers to three times the first number we get 12.Use determinants to find the numbers.

Answer 1

Given :

The sum of three numbers is 6

If we multiply the third number by 2 and add the first number the result = 7

By adding second and third numbers to three times the first number we get 12

To Find :

The numbers

Solution :

Let The number are x , y , z

The First number = x

The second number = y

The Third number = z

According to question

Statement I

The sum of three numbers is 6

i.e    x + y + z = 6                 ...........1

Statement II

If we multiply the third number by 2 and add the first number the result = 7

i.e  2 × z + x = 7

Or,    x + 2 z = 7                      ..........2

Statement III

By adding second and third numbers to three times the first number we get  12

i.e  ( y + z ) + 3 x = 12          

Or,   y + z = 12 - 3 x           ..........3

Solving eq 3 and 1

x + ( 12 - 3 x ) = 6

Or,  12 - 6 - 2 x = 0

Or,  2 x = 6

∴        x = $\dfrac{6}{2}$

i.e      x = 3

Again, Put the value of x in eq 2

Or, x + 2 z = 7

Or,   3 + 2 z = 7

Or,  2 z = 7 - 3

Or,  2 z = 4

∴        z = $\dfrac{4}{2}$

i.e      z = 2

Again, Put the value of x and z into eq 1

Or,  x + y + z = 6

Or , 3 + y + 2 = 6

Or , y + 5 = 6

∴     y = 6 - 5

i.e   y = 1

Hence, The numbers are 3 , 1 , 2     Answer