If x - y = 7 and $x^3 - y^3 = 133;$ find :

(i) xy

(ii) $x^2 + y^2$

(i) Given, x - y = 7 and $x^3 - y^3 = 133$

Using the formula :

$(x - y)^3 = x^3 - y^3 - 3xy(x - y)\\[1em]$

Substituting the values, we get:

$⇒ (7)^3 = 133 - 3xy \times 7\\[1em] ⇒ 343 = 133 - 21xy\\[1em] ⇒ 21xy = 133 - 343 \\[1em] ⇒ 21xy = -210 \\[1em] ⇒ xy = -\dfrac{210}{21}\\[1em] ⇒ xy = -10$

Hence, the value of xy = -10.

(ii) (x - y) = 7

Squaring both sides, we get:

$⇒ (x - y)^2 = 7^2\\[1em] ⇒ x^2 + y^2 - 2xy = 49\\[1em]$

From equation (i),

$⇒ x^2 + y^2 - 2 \times (-10) = 49\\[1em] ⇒ x^2 + y^2 + 20 = 49\\[1em] ⇒ x^2 + y^2 = 49 - 20\\[1em] ⇒ x^2 + y^2 = 29$

Hence, $x^2 + y^2 = 29$.