Finding the Value of x³ - y³ from Given Equations

The problem asks us to find the value of the expression $\left(x^3 - y^3\right)$ given two equations relating x and y, and a condition about their relative values:

• Equation 1: $x + y = 12$
• Equation 2: $xy = 27$
• Condition: $x > y$

To find the value of $\left(x^3 - y^3\right)$, we first need to determine the individual values of x and y.

Determining the Values of x and y

We are given the sum $\left(x+y\right)$ and the product $\left(xy\right)$ of two numbers, x and y. These two numbers can be found by considering a quadratic equation whose roots are x and y. A quadratic equation with roots r1 and r2 is given by $t^2 - \left(r1 + r2\right)t + \left(r1 \times r2\right) = 0$.

In our case, the roots are x and y, the sum is 12, and the product is 27. So, x and y are the roots of the quadratic equation:

$\qquad t^2 - \left(x+y\right)t + xy = 0$

Substituting the given values:

$\qquad t^2 - 12t + 27 = 0$

We can solve this quadratic equation for t by factoring. We look for two numbers that multiply to 27 and add up to -12. These numbers are -9 and -3.

So, the equation can be factored as:

$\qquad \left(t - 9\right)\left(t - 3\right) = 0$

This gives us two possible values for t:

• $t - 9 = 0 \implies t = 9$
• $t - 3 = 0 \implies t = 3$

The roots of the equation are 9 and 3. These are the values of x and y.

Now we use the condition $x > y$. Since 9 is greater than 3, we assign:

• $x = 9$
• $y = 3$

Let's quickly verify these values using the original equations:

• $x + y = 9 + 3 = 12$ (Matches the given sum)
• $xy = 9 \times 3 = 27$ (Matches the given product)
• $x > y$ (9 > 3) (Matches the given condition)

The values $x = 9$ and $y = 3$ are correct.

Calculating the Value of x³ - y³

Now that we have the values of x and y, we can calculate $x^3 - y^3$.

First, calculate $x^3$:

$\qquad x^3 = 9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729$

Next, calculate $y^3$:

$\qquad y^3 = 3^3 = 3 \times 3 \times 3 = 9 \times 3 = 27$

Finally, calculate the difference $x^3 - y^3$:

$\qquad x^3 - y^3 = 729 - 27 = 702$

Alternatively, we could use the algebraic identity $\left(a^3 - b^3\right) = \left(a - b\right)\left(a^2 + ab + b^2\right)$.

We already know $xy = 27$. We need $x - y$ and $x^2 + y^2$.

We can find $x - y$ using the identity $\left(x - y\right)^2 = \left(x + y\right)^2 - 4xy$:

$\qquad \left(x - y\right)^2 = \left(12\right)^2 - 4\left(27\right)$

$\qquad \left(x - y\right)^2 = 144 - 108$

$\qquad \left(x - y\right)^2 = 36$

Since $x > y$, $x - y$ must be positive:

$\qquad x - y = \sqrt{36} = 6$

Now, we need $x^2 + y^2$. We can find this using the identity $\left(x + y\right)^2 = x^2 + 2xy + y^2$:

$\qquad x^2 + y^2 = \left(x + y\right)^2 - 2xy$

$\qquad x^2 + y^2 = \left(12\right)^2 - 2\left(27\right)$

$\qquad x^2 + y^2 = 144 - 54$

$\qquad x^2 + y^2 = 90$

Now substitute these values into the identity for $x^3 - y^3$:

$\qquad x^3 - y^3 = \left(x - y\right)\left(x^2 + xy + y^2\right)$

$\qquad x^3 - y^3 = \left(6\right)\left(90 + 27\right)$

$\qquad x^3 - y^3 = \left(6\right)\left(117\right)$

$\qquad x^3 - y^3 = 702$

Both methods yield the same result, 702.

Revision Table: Key Concepts in Algebra

Additional Information: Algebraic Identities and Solving Systems

Algebraic identities are powerful tools that help simplify expressions and solve equations efficiently. In this problem, using identities like the difference of cubes or the square of a sum/difference provides alternative ways to reach the solution or verify results obtained by finding individual values of x and y.

When given the sum and product of two variables, they are the roots of a quadratic equation. Solving this quadratic equation is a standard technique to find the values of the variables. Remember to consider any additional conditions, like $x > y$ in this case, when assigning the values to the variables.

Commonly used algebraic identities include:

• $\left(a+b\right)^2 = a^2 + 2ab + b^2$
• $\left(a-b\right)^2 = a^2 - 2ab + b^2$
• $\left(a+b\right)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + b^3 + 3ab(a+b)$
• $\left(a-b\right)^3 = a^3 - 3a^2b + 3ab^2 - b^3 = a^3 - b^3 - 3ab(a-b)$
• $a^2 - b^2 = (a-b)(a+b)$
• $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
• $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

These identities are fundamental in simplifying and manipulating algebraic expressions, especially in problems involving powers of variables.