Question

Question:−
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Solve the value for x and y =?
(a-b)x+(a+b)y=a²-2ab+b²
(a+b)(x+y)=a²-b²
Please answer in good content
wrong answer will be deleted at the spot.​

Answer 1

Answer:

a+b , y = - 2ab /(a+b)

Step-by-step explanation:

(a-b) x + (a + b) y = a² - 2ab - b²…………(1)

(a + b) (x + y) = a² + b²……………………(2)

Equation (2) can be written as (a + b)x+ (a+b) y = a² + b²………….(3)

Now we have to solve equation (1) and (3)

(a-b) x + (a + b) y = a² - 2ab - b²,…………(1)

(a + b)x+ (a+b) y = a² + b²………….(3)

Subtracting equation (3) from (1) we get

(a-b-a-b) x = a² - 2ab - b²- a² - b²

-2b x = - 2ab -2 b²

-2b x =-2b(a+b)

dividing both sides by -2b

x = a+b

Now substitude x=a+b in equation (1) we get

(a-b)(a+b) + (a + b) y = a² - 2ab - b²

a² - b² + + (a + b) y = a² - 2ab - b²

Subtracting a² - b² from both the sides

(a + b) y = a² - 2ab - b² - a² + b²

(a + b) y = - 2ab

y = - 2ab /(a+b)

Answer x = a+b , y = - 2ab /(a+b)

Answer 2

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Question\;:-}}}$

Here the concept of Linear Equation in Two Variables has been used. We see we are given the two equations. Firstly we can simplify them into simplest form. Then we shall use Elimination Method to solve these equations since these are Linear Equations in Two Variables.

Let's do it !!

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★ Correct Question :-

Solve the value for x and y = ?

• (a - b)x + (a + b)y = a² - 2ab - b²

• (a + b)(x + y) = a² - b²

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★ Solution :-

Given,

$\\\;\bf{\rightarrow\;\;\green{(a\:-\:b)x\;+\;(a\:+\:b)y\;=\;a^{2}\;-\;2ab\;-\;b^{2}}}$

Let this be equation a) .

Now we are given that,

$\\\;\sf{\rightarrow\;\;(a\:+\:b)(x\:+\:y)\;=\;a^{2}\;+\;b^{2}}$

On simplifying this equation, we get

$\\\;\sf{\rightarrow\;\;ax\;+\;ay\;+\;bx\;+\;by\;=\;a^{2}\;+\;b^{2}}$

$\\\;\sf{\rightarrow\;\;ax\;+\;bx\;+\;ay\;+\;by\;=\;a^{2}\;+\;b^{2}}$

$\\\;\bf{\rightarrow\;\;\green{(a\:+\:b)x\;+\;(a\:+\:b)y\;=\;a^{2}\;+\;b^{2}}}$

Let this be equation b) .

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~ For the value of x ::

Now subtracting equation b) from equation a), we get

$\\\;\bf{\Longrightarrow\;\;\red{(a\:-\:b)x\;+\;(a\:+\:b)y\;-\;[(a\:+\:b)x\;+\;(a\:+\:b)y]\;=\;a^{2}\;-\;2ab\;-\;b^{2}\;-\;[a^{2}\;+\;b^{2}]}}$

$\\\;\sf{\Longrightarrow\;\;(a\:-\:b)x\;+\;(a\:+\:b)y\;-\;(a\:+\:b)x\;-\;(a\:+\:b)y\;=\;\bf{a^{2}\;-\;2ab\;-\;b^{2}\;-\;a^{2}\;-\;b^{2}}}$

Cancelling the like terms, we get

$\\\;\sf{\Longrightarrow\;\;(a\:-\:b)x\;-\;(a\:+\:b)x\;=\;\bf{-\;2ab\;-\;b^{2}\;-\;b^{2}}}$

$\\\;\sf{\Longrightarrow\;\;ax\:-\:bx\;-\;(ax\:+\:bx)\;=\;\bf{-\;2ab\;-\;2b^{2}}}$

$\\\;\sf{\Longrightarrow\;\;ax\:-\:bx\;-\;ax\:-\:bx\;=\;\bf{-\;2ab\;-\;2b^{2}}}$

$\\\;\sf{\Longrightarrow\;\;-\:bx\:-\:bx\;=\;\bf{-\;2ab\;-\;2b^{2}}}$

$\\\;\sf{\Longrightarrow\;\;-\;2b\:x\;=\;\bf{-\;2b(a\;+\;b)}}$

Cancelling -2b from both sides, we get

$\\\;\large{\bf{\orange{\Longrightarrow\;\;x\;=\;\bf{(a\;+\;b)}}}}$

Hence, we got the value of x.

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~ For the value of y ::

Now using the value of x and equation a) , we get,

$\\\;\bf{\rightarrow\;\;(a\:-\:b)x\;+\;(a\:+\:b)y\;=\;a^{2}\;-\;2ab\;-\;b^{2}}$

By applying the value of x,

$\\\;\sf{\Longrightarrow\;\;(a\:-\:b)(a\:+\:b)\;+\;(a\:+\:b)y\;=\;\bf{a^{2}\;-\;2ab\;-\;b^{2}}}$

$\\\;\sf{\Longrightarrow\;\;a^{2}\;+\;ab\;-\;ab\;-\;b^{2}\;+\;(a\:+\:b)y\;=\;\bf{a^{2}\;-\;2ab\;-\;b^{2}}}$

$\\\;\sf{\Longrightarrow\;\;a^{2}\;-\;b^{2}\;+\;(a\:+\:b)y\;=\;\bf{a^{2}\;-\;2ab\;-\;b^{2}}}$

Transposing the like terms to other side,

$\\\;\sf{\Longrightarrow\;\;(a\:+\:b)y\;=\;\bf{a^{2}\;-\;2ab\;-\;b^{2}\;-\;a^{2}\;+\;b^{2}}}$

$\\\;\sf{\Longrightarrow\;\;(a\:+\:b)y\;=\;\bf{-\;2ab}}$

$\\\;\large{\bf{\blue{\Longrightarrow\;\;y\;=\;\bf{\dfrac{(-\;2ab)}{(a\:+\:b)}}}}}$

Hence, we got our answer. Thus,

$\\\;\odot\;\;\bf{\purple{x\;=\;(a\;+\;b)}}$

$\\\;\odot\;\;\bf{\purple{y\;=\;\dfrac{(-\:2ab)}{(a\;+\;b)}}}$

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★ More to know ::

→ Types of Linear Equations :-

• Linear Equation in One Variable : These are the types of equations which have only one variable term.

• Linear Equation in Two Variable : These are the types of equations which have two variable terms and are solved as two equations.

• Linear Equation in Three Variable : These are the types of equations which have three variable terms.

→ Methods to solve Linear Equation in Two Variables :-

• Substitution Method

• Elimination Method

• Cross Multiplication Method

• Reducing the Pair Method