Lesson Find the numbers using system of equations

Find the numbers using system of equations

Problem 1

Let x be the larger of the two numbers and y be the smaller.

Then the condition gives you these two equations


x = 7y + 19,           (1)

3x = 11*(2y) + 19.     (2)


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


3*(7y + 19) = 22y + 19,    or

21y + 57 = 22y + 19  ====>  y = 57 - 19 = 38.


Then  x = 7*y + 19 = 7*38 + 19 = 285.


Answer.  The larger number is 285.  The smaller number is 38.


Check.   Check eq(1):  7*y+19 = 7*38+19 = 285 = x.             ! Correct !

         Check eq(2):  3x = 3*285 = 11*2*38+19 = 11*(2x)+19.   ! Correct !

Problem 2

Let x be the smaller and y be the greater number.


Then your system is


3x + 2y = 27,    (1)
4x - 2y = -6.    (2)


To solve it, add the equations (both sides).  You will get

3x + 4x = 27 - 6,    or

7x = 21  ====>    x = 21/7 = 3.


Then from eq(1),  2y = 27 - 3x = 27 - 3*3 = 18  and  y = 18/2 = 9.


Answer.  The smaller number is 3.  The greater number is 9.

Problem 3

Let x = the ones digit and y = the tens digit.


You are given


    x - y = 2           (1)

    x^2 - y^2 = 32      (2)



Equation (2) is the same as


    (x-y)*(x+y) = 32.    (3)


In equation (3), replace the factor (x-y) by 2, based on the given (1). You will get

    2*(x+y) = 32,

    x + y = 16.           (4)



We are at the finish line now . . . 

Add equations (1) and (2). You will get


    2x = 16 + 2 = 18,


hence,  x = 18/2 = 9.


Then  y is 2 less than 9, i.e. 7.


ANSWER.  The two-digit number is 79.


CHECK.   9^2 - 7^2 = 81 - 49 = 32.    ! Correct !

Problem 4

Let x, y and z be the 1-st, the 2-nd and the 3-rd numbers, respectively.

Then from the condition you have these three equations in 3 unknowns


 x +  y +  z = 14,        (1)
2x + 4y + 5z = 62,        (2)
4x -  y      = -1.        (3)


From eq(3), express  y = 4x +1  and substitute it into eq(1) and eq(2). You will get


 x + (4x+1)  +  z = 14    (1')
2x + 4(4x+1) + 5z = 62    (2')


Simplify and write it in the standard form


 5x +  z = 13             (1'')
18x + 5z = 58             (2'')


Congratulations !  You reduced the 3x3-system to 2x2-system, which is much easier to solve.


To solve it, apply Elimination method.  For it, multiply eq(1'') by 5  (both sides). You will get


25x + 5z = 65             (1''')
18x + 5z = 58.            (2''')


Now subtract eq((1''')  from eq(2'''). You will get


7x = 65 - 58 = 7  ====>  x = 1.


Then from eq(1'')  z = 13-5x = 13 - 5*1 = 8.


Finally,  y = 14 - x - z = 14 - 1 - 8 = 5.


Answer.  x = 1;  y = 5;  z = 8.

Problem 5

Let the numbers be x, y and z.


Then the problem says


x + y + z =  1      (1)
x - y + z = -1      (2)
x + y - z = -9      (3)


Do not be afraid.  Simply subtract equation (2) from equation (1).  The terms  "x"  and  "z"  will cancel each other and you will get

    2y     = 1 - (-1) = 2   =====>  y = 2/2 = 1.


Next subtract equation (3) from equation (1).  The terms "x" and "y" will cancel each other and you will get


       2z = 1 - (-9) = 10   =====>  z = 10/2 = 5.


As the last step, substitute  y= 1  and  z= 5 into equation (1). You will get


x + 1 + 5 = 1  =====>  x = -5.


Answer.  x= -5;  y= 1;  z= 5.

Problem 6

x + y + z = 8     (1)
x - y + z = 4     (2)
x - z = y + 2     (3)


Subtract  equation (2) from equation (1). You will get


2y = 4  ====>  y = 4/2 = 2.


Substitute y= 2 into equations  (1)  and  (3).   You will get


x + z = 6     (4)
x - z = 4     (5)


Add equations (4) and (5).  You will get


2x = 10  ====>  x = 10/2 = 5.


Substitute  x= 5  and  y= 2 into equation (1). You will get


5 + 2 + z = 8  ====>  z = 1.


Answer.  x= 5,  y= 2,  z= 1.

Problem 7

x - y = 5            (1)

 -  =          (2)


From equation (2),

 = .


Replace here x-y in the numerator by 5, based on (1).  You will get

     = ,


which implies

    xy = 24.    (3).



Now the problem is to find two positive integers with the difference 5 and the product 24.


It requires 5 seconds to guess the ANSWER mentally :  the numbers are  3 and 8.



Or, if you want to solve it algebraically, write this quadratic equation

    x*(x-5) = 24


and complete the solution by a standard way.