Question

Solve the following pair of linear equations a minus b into X + a + b into Y is equal to a square minus 2 a b minus b square and a + b into X + Y is equal to a square + b square

Answer 1

Question:

↓$(a-b)x + (a+b)y=a^2-2ab-b^2$$and\\(a+b)(x+y)=a^2+b^2$

Answer:$x=a+b\:and\:y=\frac{2b^2}{a+b}$

Step-by-step explanation:

let's begin by simplifying second equation $(a+b)(x+y)=a^2+b^2</p><p>[tex]ax+ay+bx+by=a^2+b^2\\\implies\:(a+b)x+(a+b)y=a^2+b^2$ ......equation 2  

subtract equation 2 from 1

$(a+b)x+(a+b)y=a^2+b^2\\-((a-b)x+(a+b)y=a^2-2ab-b^2)\\after\:subtracting\:,we\:get,\\\\(a+b)x-(a-b)x=a^2+b^2-(a^2-2ab-b^2)\\ax+bx-ax+bx=a^2+b^2-a^2+2ab+b^2\\\implies\: 2bx=2ab+2b^2\\\implies\:(2b)x=2b(a+b)\\\implies\:x=a+b\\  substitute\:the\:value\:of\:x\:in\:equ\:2\\\\(a+b)x+(a+b)y=a^2+b^2\\(a+b)(a+b)+(a+b)y=a^2+b^2\\\implies\:(a+b)(a+b+y)=a^2+b^2\\\implies\:a+b+y=\frac{a^2+b^2}{a+b}\\\implies\:y=\frac{a^2+b^2}{a+b}-a-b\\\implies\:y=\frac{a^2+b^2-(a-b)(a+b)}{a+b}$$\implies\:y=\frac{a^2+b^2-(a^2-b^2)}{a+b}\\\implies\:y=\frac{2b^2}{a+b}$