Question

Solve each of the following systems of equations by the method of cross-multiplication:
(a-b)x+(a+b)y = 2a²-2b²(a+b)(x+y)=4ab

Answer 1

From the solved given equations we get the values of x and y are $\frac{2ab-a^2+b^2}{b}$ and  $\frac{(a-b)[a^2+b^2]}{b(a+b)}$ respectively

Step-by-step explanation:

Given equations are $(a-b)x+(a+b)y=2a^2-2b^2$ and $(a+b)(x+y)=4ab$

To solve the given equations by Cross Multiplication method :

• The given equations becomes  $(a-b)x+(a+b)y-2a^2-2b^2=0$ and $(a+b)(x+y)-4ab=0$
• Now solving the above two equations
• For the equations $a_1x+by_1+c_1=0$ and $a_2x+b_2y+c_2=0$

The formula is $\frac{x}{b_1c_2-b_2c_1}=\frac{y}{c_1a_2-c_2a_1}=\frac{1}{a_1b_2-a_2b_1}$

a+b   $-(2a^2-2b^2)$  a-b       a+b

a+b      -4ab           a+b      a+b

Substitute the values we get

$\frac{x}{-4ab(a+b)+(a+b)(2a^2-2b^2)}=\frac{y}{-(a+b)(2a^2-2b^2)+4ab(a-b)}=\frac{1}{(a-b)(a+b)-(a+b)(a+b)}$

Now equating $\frac{x}{-4ab(a+b)+(a+b)(2a^2-2b^2)}=\frac{1}{(a-b)(a+b)-(a+b)(a+b)}$

• $\frac{x}{-4ab(a+b)+2(a+b)(a^2-b^2)}=\frac{1}{(a^2-b^2)-(a+b)^2}$
• $\frac{x}{-2(a+b)[2ab-(a^2-b^2)]}=\frac{1}{a^2-b^2-a^2-b^2-2ab}$

• $\frac{x}{-2(a+b)[2ab-a^2+b^2]}=\frac{1}{-2b^2-2ab}$
• $\frac{x}{-2(a+b)[2ab-a^2+b^2]}=\frac{1}{-2b(b+a)}$
• $x=\frac{-2(a+b)[2ab-a^2+b^2]}{-2b(a+b)}$
• $x=\frac{2ab-a^2+b^2}{b}$

Therefore the value of x is $\frac{2ab-a^2+b^2}{b}$

Now equating $\frac{y}{-(a+b)(2a^2-2b^2)+4ab(a-b)}=\frac{1}{(a-b)(a+b)-(a+b)(a+b)}$

• $\frac{y}{-2(a+b)(a^2-b^2)+4ab(a-b)}=\frac{1}{(a^2-b^2)-(a+b)^2}$
• $\frac{y}{-2(a+b)(a+b)(a-b)+4ab(a-b)}=\frac{1}{a^2-b^2-a^2-b^2-2ab}$
• $\frac{y}{-2(a-b)[(a+b)^2-2ab]}=\frac{1}{-2b^2-2ab}$
• $\frac{y}{-2(a-b)[a^2+b^2+2ab-2ab]}=\frac{1}{-2b(b+a)}$
• $\frac{y}{-2(a-b)[a^2+b^2]}=\frac{1}{-2b(a+b)}$
• $y=\frac{-2(a-b)[a^2+b^2]}{-2b(a+b)}$

• $y=\frac{(a-b)[a^2+b^2]}{b(a+b)}$

Therefore the value of y is $\frac{(a-b)[a^2+b^2]}{b(a+b)}$

From the solved given equations we get the values of x and y are $\frac{2ab-a^2+b^2}{b}$ and  $\frac{(a-b)[a^2+b^2]}{b(a+b)}$ respectively