Solving System of 3 Equations

Question

x+2y-z=6

4x+2y+z=19

(x+2y)² -z² =24

 

I used the elimination method on the first two equations, eliminating the z and getting 5x+4y=25

 

After that, I don't know how to simplify the third equation

Answer 1

The elimination method on the first two equations gets rid of the z, as you said.

  x + 2y +z = 6             [use eq1]

 4x +2y -z = 19            [use eq2]

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

  5x +4y    = 25          [eq4]

 

Now, notice that the third equation is the difference of squares -- (a²-b²) = (a+b)(a-b):

    ((x+2y) + z)((x+2y) - z) = 24       (from eq3)

     (x+2y+z)(6) = 24                    (see first equation)

     x + 2y + z = 4   

     x + 2y - z = 6           [use eq1]

 -----------------------  (elimination method; add equations)

   2x + 4y      = 10     [eq5]

   5x + 4y      = 25     [use eq4]

 ---------------------    (elimination method; subtract top from bottom)

   3x             = 15

     x             = 5

   2(5)  + 4y   = 10      [use line above and eq5]

              4y    = 0

               y    = 0

 

Checking (very important:

   From eq1, z=-1.

   Is  5 +2(0)-(-1) = 6   ? yes

   Is  4(5)+2(0)+(-1)=19   ?

           20        - 1    = 19   ? yes

   Is ((5+2(0))² -(-1)² =24   ?

           5⁵  -  1  = 24   ? yes

Answer 2

From the first equation,       

 

z = x + 2y - 6

 

 

To simplify the third equation, expand everything out.  Substitute the value of z into the third equation.

 

(x + 2y)(x + 2y) - (x + 2y - 6)(x + 2y - 6) = 24

 

x² + 4xy + 4y² - [x(x + 2y - 6) + 2y(x + 2y - 6) - 6(x + 2y - 6)] = 24

 

 

Distribute the terms inside brackets.

 

x² + 4xy + 4y² - [x² + 2xy - 6x + 2xy + 4y² - 12y - 6x - 12y + 36] = 24

 

 

Distribute the negative sign outside the brackets.

 

x² + 4xy + 4y² - x² - 2xy + 6x - 2xy - 4y² + 12y + 6x + 12y - 36 = 24

 

 

Combine like terms and keep the constant terms one side of the equation.

 

 

12x + 24y = 60          new eq3

 

 

Now you have only two equations.

 

5x + 4y = 25

12x + 24y = 60