Question

If $3x - 2y = 13$ and $xy = 15$, then find the value of $27{x^3} - 8{y^3}$.

Answer 1

Hint: We are required to find the value of the given equation provided the conditions of the variables involved. So, we will use the algebraic identity $(a - b)^3$ to solve the question and get the required value.

Complete step-by-step answer:
We need to find the value of the given equation: $27{x^3} - 8{y^3}$
We are given that $3x - 2y = 13$and $xy = 15$
On cubing both sides of the equation $3x - 2y = 13$, we get
${\left( {3x - 2y} \right)^3} = \left( {{{13}^3}} \right)$ …eq (1)
Using the algebraic identity ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)$ , we get
$ \Rightarrow {\left( {3x - 2y} \right)^3} = {\left( {3x} \right)^3} - {\left( {2y} \right)^3} - 3\left( {3x} \right)\left( {2y} \right)\left( {3x - 2y} \right)$
Now, we can substitute the above value in eq (1). On substituting the value, we get
$ \Rightarrow $${\left( {3x} \right)^3} - {\left( {2y} \right)^3} - 3\left( {3x} \right)\left( {2y} \right)\left( {3x - 2y} \right) = {\left( {13} \right)^3}$
$ \Rightarrow 27{x^3} - 8{y^3} - 18xy\left( {3x - 2y} \right) = 2197$
We were given, initially, the values of: $3x - 2y = 13$, $xy = 15$. Substituting these values in the above equation, we get:
$ \Rightarrow 27{x^3} - 8{y^3} - 18\left( 5 \right)\left( {13} \right) = 2197$
$ \Rightarrow 27{x^3} - 8{y^3} - 1170 = 2197$
$ \Rightarrow 27{x^3} - 8{y^3} = 3367$
Hence, the value of the required equation $27{x^3} - 8{y^3}$ is 3367.

Note: In such problems, we generally take a look at the given initial conditions and solve accordingly. We can also use various properties as there can be many ways to solve a question. We can also solve this question using the identity $a^3$ – $b^3$ in the equation $27{x^3} - 8{y^3}$ and then upon expanding we will get our desired answer. By substitution also, we can find the value of $27{x^3} - 8{y^3}$.
We used an algebraic identity in the above mentioned question. We can define the algebraic identities as: an equality which holds for each and every value of its variables is called an algebraic identity or an identity is an equality relating one mathematical expression with the other.
In other words, we can say that an algebraic identity is basically an equality in which every value of its variables holds true.