if 3x + 2y = 5 and xy = - 2 , find 27x³ + 8y³

We are given two equations:

1. \( 3x + 2y = 5 \)
2. \( xy = -2 \)

We need to find the value of \( 27x^3 + 8y^3 \).

### Step-by-Step Solution:

#### Step 1: Recognize the form of the expression.
The expression \( 27x^3 + 8y^3 \) can be rewritten as:

\[
27x^3 + 8y^3 = (3x)^3 + (2y)^3
\]

#### Step 2: Use the identity for sum of cubes.
We use the identity for the sum of cubes:

\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

Here, let \( a = 3x \) and \( b = 2y \). So we have:

\[
(3x)^3 + (2y)^3 = (3x + 2y)((3x)^2 - (3x)(2y) + (2y)^2)
\]

#### Step 3: Substitute \( 3x + 2y \) and simplify.
From the first equation, we know that \( 3x + 2y = 5 \). Substituting this, we get:

\[
(3x)^3 + (2y)^3 = 5 \left( (3x)^2 - (3x)(2y) + (2y)^2 \right)
\]

Now, we calculate each term inside the parentheses:

- \( (3x)^2 = 

if 3x + 2y = 5 and xy = -2, find 27x³ + 8y³

Explore how to solve the algebraic problem where 3x + 2y = 5 and xy = -2, and find the value of 27x³ + 8y³ using the sum of cubes identity. This solution involves key concepts in algebra, suitable for students in Grades 9-11.

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