Question

If 3x – 4y = 5 and xy = 3, then find $27{{\text{x}}^3} - 64{{\text{y}}^3}$.

$
  {\text{A}}{\text{. 125}} \\
  {\text{B}}{\text{. 665}} \\
  {\text{C}}{\text{. 225}} \\
  {\text{D}}{\text{. 985}} \\
$

Answer 1

Hint: To compute the given term, we use ${{\text{a}}^3} - {{\text{b}}^3}$ formula and expand the term $27{{\text{x}}^3} - 64{{\text{y}}^3}$ and we use the formula ${\left( {{\text{a - b}}} \right)^2}$ to obtain the required terms and simplify it.

Complete step-by-step answer:
We know,
$
  {{\text{a}}^3} - {{\text{b}}^3}{\text{ = }}\left( {{\text{a - b}}} \right)\left( {{{\text{a}}^2} + {\text{ab + }}{{\text{b}}^2}} \right) \\
  {\left( {{\text{a - b}}} \right)^2}{\text{ = }}{{\text{a}}^2} + {{\text{b}}^2} - {\text{2ab}} \\
$

Now let us simplify, $27{{\text{x}}^3} - 64{{\text{y}}^3}$ using the formula ${{\text{a}}^3} - {{\text{b}}^3}{\text{ = }}\left( {{\text{a - b}}} \right)\left( {{{\text{a}}^2} + {\text{ab + }}{{\text{b}}^2}} \right)$
Where a = 3x and b = 4y
$
   = {\left( {3{\text{x}}} \right)^3} - {\left( {{\text{4y}}} \right)^3} \\
   = \left( {{\text{3x - 4y}}} \right)\left( {{{\left( {{\text{3x}}} \right)}^2} + {\text{3x}}{\text{.4y + }}{{\left( {{\text{4y}}} \right)}^2}} \right) \\
   = \left( {{\text{3x - 4y}}} \right)\left( {{\text{9}}{{\text{x}}^2} + 12{\text{xy + 16}}{{\text{y}}^2}} \right){\text{ - - - (1)}} \\
$

Given Data, 3x – 4y = 5
Squaring on both sides of this equation,
$ \Rightarrow {\left( {{\text{3x - 4y}}} \right)^2} = {\text{ }}{{\text{5}}^2}$
It is of the form${\left( {{\text{a - b}}} \right)^2}{\text{ = }}{{\text{a}}^2} + {{\text{b}}^2} - {\text{2ab}}$, where a = 3x and b = 4y
$ \Rightarrow {\text{9}}{{\text{x}}^2} + 16{{\text{y}}^2} - 24{\text{xy = 25}}$
Now we add and subtract 12xy to the equation, we get
$
   \Rightarrow {\text{9}}{{\text{x}}^2} + 16{{\text{y}}^2} - 24{\text{xy + 12xy - 12xy = 25}} \\
   \Rightarrow {\text{9}}{{\text{x}}^2} + 16{{\text{y}}^2} - 36{\text{xy + 12xy = 25}} \\
   \Rightarrow {\text{9}}{{\text{x}}^2} + 16{{\text{y}}^2}{\text{ + 12xy = 25 + 36xy}} \\
$

Substituting the given Data: xy = 3 in the above equation we get,
$
   \Rightarrow {\text{9}}{{\text{x}}^2} + 16{{\text{y}}^2}{\text{ + 12xy = 25 + 36}} \times {\text{3}} \\
   \Rightarrow {\text{9}}{{\text{x}}^2} + 16{{\text{y}}^2}{\text{ + 12xy = 25 + 108}} \\
   \Rightarrow {\text{9}}{{\text{x}}^2} + 16{{\text{y}}^2}{\text{ + 12xy = 133}} \\
$

Now, substituting ${\text{9}}{{\text{x}}^2} + 16{{\text{y}}^2}{\text{ + 12xy = 133}}$ and 3x – 4y = 5 in equation (1), we get
⟹$\left( {{\text{3x - 4y}}} \right)\left( {{\text{9}}{{\text{x}}^2} + 12{\text{xy + 16}}{{\text{y}}^2}} \right)$
$
   \Rightarrow 27{{\text{x}}^3} - 64{{\text{y}}^3} = {\text{ 5 }} \times {\text{ 133}} \\
   \Rightarrow 27{{\text{x}}^3} - 64{{\text{y}}^3}{\text{ = 665}} \\
$

Hence Option B is the correct answer.

Note: In order to solve questions of this type the key is to identify the nature of the given question and use the formula that fits and we use it accordingly. Having adequate knowledge in algebraic formulae is essential, i.e. in this case that is, ${{\text{a}}^3} - {{\text{b}}^3}$ and then using ${\left( {{\text{a - b}}} \right)^2}$ to get the required terms.