Question

If 3x-2y=5 and xy=6 find the value of 27x^3-8y^3

Answer 1

First, use the simultaneous equation method to find x and y by letting the first equation be 3x - 2y = 5 and the second equation be xy = 6.

3x - 2y = 5 (1st equation)

xy = 6 (2nd equation)

In the 2nd equation, move y over to 6 by dividing.

x = $\frac{6}{y}$

Then, substitute x into the 1st equation.

3($\frac{6}{y}$) - 2y = 5

Move 2y over to 5.

3($\frac{6}{y}$) = 5 + 2y

Multiply 3 with the numerator of the fraction, 6.

$\frac{18}{y}$ = 5 + 2y

Move the y on the left side of the equation to the right and multiply.

18 = y(5 + 2y)

18 = 5y + 2y²

Leave the equation in general form by letting the values equal to 0.

2y² + 5y - 18 = 0

Factorise the equation.

(y - 2)(2y + 9) = 0

Solve for the values of y.

y - 2 = 0

y = 2

2y + 9 = 0

2y = -9

y = -9/2

Substitute the values of y into the 2nd equation and solve for x.

x(2) = 6

2x = 6

x = 3

x(-9/2) = 6

x = 6 ÷ -9/2

x = 6 × -2/9

x = -4/3

Now, we will find the values of 27x³ - 8y³.

First, substitute the known values of x and y into the equation.

• 27(3)³ - 8(2)³
• 27(-4/3)³ - 8(-9/2)³

Find each of the values to the power of 3.

• 27(-27) - 8(-8)
• 27(-64/27) - 8(-729/8)

Multiply 27 and 8 with the numbers in the brackets beside them.

• -729 + 64
• -64 + 729

Add the numbers.

• -665
• 665

Therefore, the values are -665 and 665.

Answer 2

$\huge{\pink{\bold{\underline{\ulcorner{\star\:Solution\: \star}\urcorner}}}}$

$\red{\underline{Given}} \colon \\\\ 3x-2y= 5\\xy=6$  

$\red{\underline{Solution\colon}}\\ We\: know \: \\\\a^{3}\: - \:b^{3} = (a-b)(a^{2}+ab+b^{2})\\\\ Now,\\\\ 27x^{3}-8y^{3} = (3x)^{3}-(2y)^{3} \\ where\: a = 3x \: and \:b= 2y$

=> (3x-2y){(3x)²+(2y)²+(3x.2y)}

[*(a-b)² = a²+b²-2ab

*(a-b)²+2ab = a²+b²-2ab+2ab

*(a-b)²+2ab= a²+b²]

=> (3x-2y){(3x-2y)²+2×3x×2y+(6xy)}

=> (3x-2y){(3x-2y)²+12xy+6xy}

=> (3x-2y){(3x-2y)²+18xy}

put the values in equation :

=> (5){(5)²+18×6}

=> 5{25+108}

=> 5{133}

=> 665

$\red{\bold{\underline{Answer}}}$

$\green{\boxed{value\: = \: 665}}$