Question

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Answer 1

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Answer 2

Answer:

The value of $x^3+125y^3+150xy-1000$ will be equal to 0.

Solution:

From the given information, we know that,

$x+5y=10\\\\5y=10-x$

Given,

$\begin{array} { c } { x ^ { 3 } + 125 y ^ { 3 } + 150 x y - 1000 } \\\\ { = x ^ { 3 } + 125 y ^ { 3 } + 5 \times 30 x y - 1000 } \end{array}$

Splitting out the terms in term of 5y and by substituting $5y=10-x$ in the above expression, we get

$\begin{array} { c } { = x ^ { 3 } + 125 y ^ { 3 } + 30 x ( 10 - x ) - 1000 } \\\\ { = x ^ { 3 } + 125 y ^ { 3 } + 300 x - 30 x ^ { 2 } - 1000 } \\\\ { = x ^ { 3 } + 5 \times 5 \times 5 y ^ { 3 } + 300 x - 30 x ^ { 2 } - 1000 } \\\\ { = x ^ { 3 } + ( 10 - x ) ^ { 3 } + 300 x - 30 x ^ { 2 } - 1000 } \end{array}$

We know that, $(a-b)^3=(a-b)(a^2+b^2-2ab)$

Substituting this in the above expression, we get

$\begin{array} { c } { = x ^ { 3 } + ( 10 - x ) \left( 10 ^ { 2 } + x ^ { 2 } - 20 x \right) + 300 x - 30 x ^ { 2 } - 1000 } \\\\ { = x ^ { 3 } + 1000 + 10 x ^ { 2 } - 200 x - 100 x - x ^ { 3 } + 20 x ^ { 2 } + 300 x - 30 x ^ { 2 } - 1000 } \\\\ { = x ^ { 3 } - x ^ { 3 } + 1000 - 1000 + 30 x ^ { 2 } - 30 x ^ { 2 } - 300 x + 300 x } \\\\ { = 0 } \end{array}$

Therefore, the value of $x^3+125y^3+150xy-1000$ will be equal to 0.