Question

40. If x - y = 7 and x3 – y3 = 133; find :
(i) xy
(ii) x2 + y2

Answer 1

hope it will help you:-)

Answer 2

Answer:

hence xy = -10  and x^2 + y^2 = 29

Step-by-step explanation:

Solving the equation

x - y = 7 and x^3 – y^3 = 133

we can solve this by using substitution method

x=7+y

now place this into the formula x^3 – y^3 = 133 where x is:

(7 +y)^3 - y^3 = 133

( 7 +y)^3

can be solved like this:

$(a + b)^{3} =a^{3} + 3a^{2} b +3a b^{2} + b^{3} \\\\a = 7 , b = y\\\\( 7+ y)^{3} =7^{3} + 3*7^{2} b +3*7 *y^{2} + y^{3} \\( 7+ y)^{3} = 343 +147y+ 21y^{2} + y^{3} \\\\= y^{3} + 21y^{2} +147y +343$

$y^{3} + 21y^{2} + 147y +343 - y^{3} = 133$

both the y^3 cancel out

$21y^{2} +147y +343 = 133\\21y^{2} +147y + 343 - 133 = 0\\21y^{2} +147y + 210 = 0$

we can then simplify this

the common factor is 21

$\frac{21y^{2} }{21} + \frac{147y}{21} +\frac{210}{21}$

= $y^{2} +7y +10 = 0$

we can now use product and sum to get the value of y

product is 10

sum is 7

hence common number use to achieve these values are:

5 and 2

this can be justified as :

5*2 = 10 and 5+2 = 7

this becomes :

(y +5) + (y+2) = 0

hence y = -5 and y = -2

to get x we put the values of y into the original formula: x - y = 7

first lets put y = -5

x - (-5) = 7

x + 5 = 7

x = 7-5

x = 2

x - y = 7

lets put y = -2

x - (-2) = 7

x + 2 = 7

x = 7-2

x = 5

lets test if this is correct with the second formula given : x^3 – y^3 = 133

if we put y = -5 and x = 2

$2^{3} - (-5)^{3}= \\8 + 125 \\= 133\\correct$

lets also test if y = -2 and x = 5

$5^{3}+ (-2)^{3} =\\125 +8 \\= 133\\correct$

we can say that:

x = 5 and y = -2   AND    x = 2 and y = -5

solve for (i) xy

if you put x = 5 and y = -2 or x = 2 and y = -5

then :

5*-2 = -10 and 2*-5 = -10

both answers = -10

hence xy = -10

solve for (ii) x^2 + y^2

if you put  x = 5 and y = -2 or x =2  and y = -5

then :

5^2 +(-2) ^2 = 29  and 2^2 + (-5)^2 = 29

both answers = 29

hence x^2 + y^2 = 29