Solve the linear equation

If 3x + 2y = 16 and 2x + 3y = 19 , then x + y = ...................

Solution :    3x + 2y = 16

                     2x + 3y = 19.           ( add) 

                      ....................

                      5x + 5y = 35

                        x + y.   = 7.            ( remove 5

                                                         common) 

         

Linear equation. Digit sum.

If the digit in the unit place of a two digit number is 2x and the digit in the ten's place is y.  Then the two digit number is..............

Solution :  Ten's.        Unit

                      y.             2x

                   x 10.          x 1

                  ........................

                  10y.    +     2x

Therefore the number is 2x + 10y. 

Five years hence , the age of Jacob will bethree times that of his son. Five years ago Jacob's age was seven times that of his son. What are their present ages

Solution  :

Let the present age of Jacob be x

Let the present age of his son be y

After five years 

Jacob will be ( x + 5 )

His son will be ( y + 5 )

According to the first statement 

x + 5 = 3 ( y + 5 )

x + 5 = 3y + 15

x - 3y = 15 - 5

                            x - 3y = 10.         Equation ( 1 )

Five years ago

Jacob will be ( x - 5 )

His son's  age will be ( y - 5)

According to the second statement 

x - 5 = 7 ( y - 5 )

x - 5 = 7y - 35

x - 7y = -35 + 5

x - 7y = - 30.           Equation ( 2 )

Take both equations and solve

x - 3y = 10

x - 7y = -30

                                     -   +        +                  ( change  signs) 

...................

     +4y = + 40

          y = 10

Substitute y =10 in equation  ( 1 )

x - 3y = 10

x - 3 ( 10 ) = 10

x -  30 = 10

x = 10 +30

x = 40

Hence  age of Jacob is 40 years

And his son's age is 10 years. 

   

 

MCQ for entrance

Five years ago nuri was thrice as old as Sonu.Ten years later,Nuri will be twice as old as sonu.How old are Nuri and Sonu now

Step 1: identify the two variables 

             Nuri = x

            Sonu = y

Step 2 : formulate the equations

Five years ago nuri was thrice as old as Sonu

 

                         x - 5 = 3(y - 5)

                         x - 5 = 3y - 15

                         x - 3y = -15 + 5

Thus, Equation 1 is          

                         x - 3y = - 10

Ten years later,Nuri will be twice as old as sonu

                        x + 10 = 2(y + 10)

                        x + 10 = 2y + 20

                        x - 2y = -10 + 20

Thus, Equation 2 is

                       x - 2y = 10

Step 3 : Eleminate one variable

Subtracting both Equations

                       x - 3y = -10 -(x - 2y  = 10)

                       x - 3y - x + 2y = -10 -10

                                     y = 20

Step 4: Substitute the value of y in Equation 1

                       x - 3y = - 10

                       x - 3(20) = - 10

                       x - 60 = - 10

                       x = - 10 + 60 

                       x = 50

Therefore Nuri is 50 years old now and Sonu is 20 years old

A two digit number is four times the sum of its digits.If 18 is added to the number, the new number obtained is that by interchanging the digits of the original number.Find the number

Step1: 

A two digit number implies one number in the tens place and one number in the units/ ones place

                   Tens.     Units

                     10x        y

From above we consider the digit in the tens place as x and the digit in the units placa as y

Step 2 : 

A two digit number is four times the sum of its digits

                     10x + y = 4(x + y)

                     10x + y = 4x + 4y

                     10x - 4x + y - 4y = 0

                      6x - 3y = 0

                      3(2x - y) = 0

                         2x - y = 0/3

Hence Equation 1 is

                         2x - y = 0

If 18 is added to the number, the new number obtained is that by interchanging the digits of the original number

 

By interchanging the digits we get

                        Tens        Units

                          10y           x

Thus,

                       (10x + y) +18 = 10y + x

                         10x - x + y - 10y = -18

                          9x - 9y = -18

                           9(x - y = -2)

Thus Equation 2 is

                            x - y = -2

Step 3: Elemination of y

 

Subtracting both the Equations 

                      (2x - y = 0) - (x - y = -2)

                       2x - y - x + y = 2

                                 x = 2

Step 4: substituting the value of x in Equation 1

                          2x - y = 0

                          2x = y 

                          2(2) = y

                           y = 4

Thus the number is 

                   10x + y= 10(2) + 4 = 20 + 4 = 24

                      

Two years ago,a father was five times the age of his son.Two years later his age will be eight more than three times the age of his son.Find the present age of father and son

Step 1: identify the two variables 

             Father = x

              Son = y

Step 2 : formulate the equations

Two years ago,a father was five times the age of his son

 

                         x - 2 = 5(y - 2)

                         x - 2 = 5y - 10

                         x - 5y = -10 + 2

Thus, Equation 1 is          

                         x - 5y = - 8

Two years later his age will be eight more than three times the age of his son 

                        x + 2 = 3(y + 2) + 8

                        x + 2 = 3y + 6 + 8

                        x + 2 = 3y + 14

                        x - 3y = 14 - 2 

Thus, Equation 2 is

                       x - 3y = 12

Step 3 : Eleminate one variable

Subtracting both Equations

                       x - 5y = - 8 - (x - 3y = 12)

                       x - x -5y + 3y = - 8 - 12

                            -2y = -20

                               y = (-20) / (-2)

                               y = 10

Step 4: Substitute the value of y in Equation 1

                       x - 5y = - 8

                       x - 5(10) = - 8

                       x - 50 = - 8

                       x = - 8 + 50 

                       x = 42

Therefore father is 42 years old now and son is 10 years old

Five books and seven pens together cost Rs 79 whereas seven books and five books together costs Rs77,Find the cost of one book and two pens

Solution:

Step 1: Identify the articles

            We Identify two articles 

             1. Book

             2. Pen

Step 2: Assign variables to the articles

             Book = x

             Pen    = y

Step 3: Deduce the problem into equations

             Five books and seven pens together cost Rs 79         ---->   5x +7y = 79

             seven books and five books together costs Rs77       ---->   7x +5y = 77 

Step 4: Find one variable by eliminating the other

              Equation 1                      5x +7y = 79        

              Equation 2                      7x +5y = 77 

              Multiply Equation 1 with coefficient of x from Equation 2

               

                                                      ( 5x +7y = 79 ) * 7

                                                      35x + 49y = 553

            Multiply Equation 2 with coefficient of x from Equation 1

                                                      ( 7x +5y = 77 ) * 5

                                                      35x +25y = 385

Step 5: Subtract the two new Equations

                            

                                                          35x + 49y = 553 - (35x +25y = 385)

                                                                    (35 - 35)x + (49 - 25)y = (553 - 385)

                                                                                                  24y = 168   

                                                                                                      y = 168 / 24

                                                                                                      y = 7

            So we can say that each Pen cost Rs 7    

Step 6: Substitute the value of y in Equation 1 to Find x

                                                                   5x +7y = 79

                                                                5x + 7(7) = 79

                                                                    5x +49 = 79  

                                                                           5x = 79 - 49   

                                                                           5x = 30

                                                                             x = 30/5

                                                                             x = 6   

             So we can say that each Book cost Rs 6                

        

                                       

             Now we need to find cost for cost of one book and two pens

                                                            6 + 2(7) = 6 +14 = 20

            Cost ofone book and two pens is Rs 20