Question

7. If x + y = 7 and x - y = 3, evaluate
(a) xy. (b) x^2 + y^2​

Answer 1

Given :-

• x + y = 7
• x - y = 3

To find :-

• xy and x² + y²

Solution :-

Given two equations From these equations Lets find values of x, y

x + y = 7

x - y = 3

Adding two equations

x + y + x - y = 7 + 3

x + x + y - y = 10

2x = 10

x =10/2

x =5

Substitute value of x in any equation

x + y = 7

5 + y = 7

y = 7 - 5

y = 2

So, the values of x, y are 5,2

Now finding xy, x² + y²

xy = 5(2)

xy = 10

x² + y² = (5)² + (2)²

x² + y² = 25 + 4

x² + y² = 29

So, xy = 10 & x² + y² = 29

_____________________

Know more some algebraic identities:-

(a+ b)² = a² + b² + 2ab

( a - b )² = a² + b² - 2ab

( a + b )² + ( a - b)² = 2a² + 2b²

( a + b )² - ( a - b)² = 4ab

( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca

a² + b² = ( a + b)² - 2ab

(a + b )³ = a³ + b³ + 3ab ( a + b)

( a - b)³ = a³ - b³ - 3ab ( a - b)

If a + b + c = 0 then a³ + b³ + c³ = 3abc

Answer 2

$\huge\sf\underline{Answer}:$
$\large\tt \implies a) xy = 10 \\ \large \tt \implies b) x^2 + y^2 = 29$

$\huge\sf\underline{Solution}:$
$\large\sf\underline{Given }:$
$\: \: \: \: \: \: \: \: \: x + y = 7....(i) \\ \: \: \: \: \: \: \: \: \: x - y = 3....(ii)$

$\large\sf\underline{To \: Find }:$
$\: \: \: \: \: \: \: \: \: a) xy \\ \: \: \: \: \: \: \: \: \: b) x^2+y^2$

Let's do it,

⟼ x + y = 7 ....(i)
$\implies \boxed{\mathfrak x = 7-y} ....(iii)$
$\\ \sf \colorbox{skyblue}{\sf Substitute \: x \: in \: (ii) equation}$
⟼ x - y = 3 ....(ii)
$\implies \sf 7-y -y =3 \\ \implies \sf 7 -2y = 3 \\ \implies \sf -2y = -4 \\ \implies\color{red} \underline{\boxed{\mathcal y = 2} }$
$\\ \sf \colorbox{skyblue}{\sf Now, put \: the\: value \: of y \: (i) equation}$

⟼ x + y = 7 ....(i)
$\implies\sf x + 2 = 7 \\ \implies\color{red} \underline{\boxed{\mathfrak x = 5} }$

we have got the values of x and y,
$\large a) xy \\ \implies (5)(2) \\ \large \implies \underbrace {10}$

$\large (b) x^{2} + y^{2​} \\ \implies 5^{2} + 2^{2} \\ \implies 25 + 4 \\ \implies \underbrace{29}$

Therefore,
● xy = 10
● x² + y² = 29

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