Question

Solve for x and y by cross multiplication method: x + y= a + b , ax – by= a² - b².

Answer 1

CROSS - MULTIPLICATION METHOD:

The general form of a pair of linear equations

a1x + b1y + c1 = 0 , &        a2x + b2y + c2 = 0.

When a1 / a2 ≠ b1 / b2, the pair of linear equations will have a unique solution.
To solve this pair of equations for x and y using cross-multiplication, we’ll arrange the variables x and y and their coefficients a1, a2, b1 and b2, and the constants c1 and c2  as shown below

⇒ x =  b1 c2 - b2 c1  / a1 b2 - a2 b1   
⇒ y =  c1 a2 - c2 a1  / a1 b2 - a2 b1

The above equation is generally written as :
x/ b1 c2 - b2 c1 =  y/ c1 a2 - c2 a1 = 1/a1 b2 - a2 b1

SOLUTION:
x + y= a + b , ax – by= a² - b².

The given system of equations may be written as:
x + y-( a + b)= 0 , ax – by -( a² - b²)= 0.

a1= 1, b1= 1, c1= -(a+b)
a2= a , b2= -b , c2= -(a²-b²)

    x                       y                        1
-----------   =   -----------------     =    ---------
b1      c1        c1           a1           a1        b1

b2      c2       c2           a2           a2       b2
    x                     y                    1
-----------------    ------------- =   ---------
1    - a-b              -a-b     1         1    1

-b   -a²+b²           -a²+b²   a        a   -b
   x                            y
---------------------- =  ----------------------- =
-(a²-b²)-(-b)×-(a+b) (-a-b)×a -(-a²+b²)×1
  1
-----------
1×-b -a×1
    x                     y                            1
----------- =     ---------------  =    -----------------
(-a²+b²)-ab-b²  -a²-ab+a²-b²        -b -a
  x                 y               1
---------   = --------  =     ----------
-a²-ab        -ab - b²         -a - b
   x                 y               1
---------   = --------  =     ----------
-a(a+b)     -b(a + b)        -(a + b)

x= -a(a+b) /-(a + b)
x= a

y = -b(a + b) / -(a + b)
y= b

Hence , the solution of the given system of equation is x= a , y= b

HOPE THIS WILL HELP YOU...

Answer 2

Step-by-step explanation:

Your answer will be

X= a

Y= b