Question

Solve each of the following systems of equations by the method of cross-multiplication:
(x/a) + (y/b) = 2
ax – by = a² – b²

Answer 1

The values of x and y in the given system of equations is a and b respectively.

Step-by-step explanation:

Given equations are $\frac{x}{a}+\frac{y}{b}=2\hfill (1)$ and

$ax-by=a^2-b^2\hfill (2)$

To solve the given equations by Cross Multiplication Method :

• Given equations can be written as
• $\frac{x}{a}+\frac{y}{b}-2=0\hfill (1)$ and

$ax-by-a^2+b^2=0\hfill (2)$

By Cross Multiplication Method we have

a        -2ab            b     a

-a      $-a^2+b^2$        a    -b

• We may write it as

• $\frac{x}{-a^3+ab^2-2ab^2}=\frac{y}{-2a^b+a^2b-b^3}=\frac{1}{-b^2-a^2}$

• Now equating $\frac{x}{-a^3+ab^2-2ab^2}=\frac{1}{-b^2-a^2}$
• $x=\frac{1(-a^3+ab^2-2ab^2)}{-b^2-a^2}$

• $x=\frac{-a^3-ab^2}{-b^2-a^2}$

• $=\frac{a(-a^2-b^2)}{-b^2-a^2}$
• $=\frac{a(-a^2-b^2)}{-a^2-b^2}$

• $x=a$

• Now equating $\frac{y}{-2a^2b+a^2b-b^3}=\frac{1}{-b^2-a^2}$
• $y=\frac{1(-2a^2b+a^2b-b^3)}{-b^2-a^2}$

• $y=\frac{-a^2b-b^3}{-b^2-a^2}$

• $=\frac{b(-a^2-b^2)}{-b^2-a^2}$

• $=\frac{b(-a^2-b^2)}{-a^2-b^2}$

• $y=b$

Therefore the values of x and y is a and b respectively

Answer 2

The value of x is a ,

The value of y is b  

Step-by-step explanation:

Given as :

The system of equation are

$\dfrac{x}{a} + \dfrac{y}{b}$  = 2                      

Applying cross multiplication

$\dfrac{bx + ay}{ab}$  = 2

i.e  b x + a y = 2 ab                       ..........1

Similarly

a x - b y = a² - b²                           ..........2

Solving eq 1 and eq 2

b × ( b x + a y ) + a × ( a x - b y ) =  b × 2 a b + a × ( a² - b² )

Or, b² x + a b y + a² x - b a y   = 2 b² a  + a³ - ab²

Or, x ( a² + b² ) + y ( ab - ab ) = a³ + ab²

Or, x ( a² + b² ) + 0 = a x ( a² + b² )

∴    x = $\dfrac{a (a^{2}+b^{2}) }{(a^{2}+b^{2}) }$

i.e  x = a

So, The value of x = a

Put the value of a in eq 1

b x + a y = 2 ab

i.e  b × a + a y = 2 ab

Or, a y = 2 ab - ab

Or, a y =  ab

∴    y = $\dfrac{ab}{a}$

i.e  y = b

So, The value of y = b

Hence, The value of x is a , and The value of y is b     Answer