Understanding the Problem

The problem asks for the value of the expression \(x^3 + 216y^3\), given two equations involving \(x\) and \(y\): \(x + 6y = 8\) and \(xy = 2\), with the additional condition that \(x > 0\). The expression \(x^3 + 216y^3\) looks like a sum of cubes, specifically \(x^3 + (6y)^3\).

Key Algebraic Identities

To solve this problem, we can use the algebraic identity for the sum of cubes:

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

Alternatively, we can use another form derived from the cube of a sum:

\((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + b^3 + 3ab(a+b)\)

Rearranging this, we get:

\(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\)

Applying the Identity to \(x^3 + 216y^3\)

The expression we need to evaluate is \(x^3 + 216y^3\). We can write \(216y^3\) as \((6y)^3\). So, the expression is \(x^3 + (6y)^3\).

Let \(a = x\) and \(b = 6y\). Using the identity \(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\), we substitute \(a=x\) and \(b=6y\):

\(x^3 + (6y)^3 = (x + 6y)^3 - 3(x)(6y)(x + 6y)\)

This simplifies to:

\(x^3 + 216y^3 = (x + 6y)^3 - 18xy(x + 6y)\)

Using the Given Information

We are given the following values:

• \(x + 6y = 8\)
• \(xy = 2\)

Now we can substitute these values directly into the expanded expression:

\(x^3 + 216y^3 = (8)^3 - 18(2)(8)\)

Calculating the Result

Let's calculate the terms:

• \(8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512\)
• \(18 \times 2 \times 8 = 36 \times 8\)

To calculate \(36 \times 8\):

So, \(18 \times 2 \times 8 = 288\).

Now substitute these results back into the equation for \(x^3 + 216y^3\):

\(x^3 + 216y^3 = 512 - 288\)

Let's perform the subtraction:

Thus, the value of \(x^3 + 216y^3\) is \(224\).

The condition \(x > 0\) ensures that a real solution for \(x\) and \(y\) exists. If we solve \(x+6y=8\) for \(x\) (\(x=8-6y\)) and substitute into \(xy=2\), we get \((8-6y)y = 2\), which is \(8y - 6y^2 = 2\), or \(6y^2 - 8y + 2 = 0\). Dividing by 2 gives \(3y^2 - 4y + 1 = 0\). Factoring gives \((3y-1)(y-1) = 0\), so \(y = 1/3\) or \(y=1\). If \(y=1/3\), \(x = 8 - 6(1/3) = 8 - 2 = 6\). If \(y=1\), \(x = 8 - 6(1) = 8 - 6 = 2\). Both pairs \((6, 1/3)\) and \((2, 1)\) satisfy \(x+6y=8\) and \(xy=2\), and in both cases, \(x > 0\). Since the problem asks for a unique value of the expression, the result must be independent of which specific pair of \((x, y)\) values is chosen.

Conclusion

Using the algebraic identity for the sum of cubes and the given values for \(x+6y\) and \(xy\), we found the value of \(x^3 + 216y^3\).

Revision Table: Algebraic Identities

Additional Information: Solving Systems of Equations

The problem involves a system of two equations with two variables, \(x\) and \(y\):

1. \(x + 6y = 8\) (Linear equation)
2. \(xy = 2\) (Non-linear equation)

This is a system of a linear equation and a non-linear equation. Such systems can often be solved by substitution. We solved the first equation for \(x\) (or \(y\)) and substituted it into the second equation, resulting in a quadratic equation in one variable. Solving the quadratic equation gives the possible values for one variable, which can then be used to find the corresponding values for the other variable using the linear equation.

In this specific case, we found two pairs of \((x, y)\) values that satisfy the given equations: \((6, 1/3)\) and \((2, 1)\). Both pairs satisfy the condition \(x > 0\). The value of the expression \(x^3 + 216y^3\) should be the same for both pairs.

• For \((x,y) = (6, 1/3)\): \(6^3 + 216(1/3)^3 = 216 + 216(1/27) = 216 + 8 = 224\).
• For \((x,y) = (2, 1)\): \(2^3 + 216(1)^3 = 8 + 216(1) = 8 + 216 = 224\).

As expected, the value is indeed the same for both valid solutions to the system.