Calculate the value of x and y if (27x ÷ 81x+2y = 9 ,x + 4y = 0

To solve this problem, we need to handle the two given equations:

1. $\frac{27^x}{81^{x+2y}} = 9$
2. $x + 4y = 0$

Step 1: Simplify the first equation

All the bases (27, 81, and 9) are powers of 3. Let's express them as powers of 3:

• $27 = 3^3$
• $81 = 3^4$
• $9 = 3^2$

Substitute these into the first equation: $\frac{(3^3)^x}{(3^4)^{x+2y}} = 3^2$

Using the power rule $(a^m)^n = a^{mn}$: $\frac{3^{3x}}{3^{4(x+2y)}} = 3^2$ $\frac{3^{3x}}{3^{4x+8y}} = 3^2$

Using the division rule $\frac{a^m}{a^n} = a^{m-n}$: $3^{3x - (4x + 8y)} = 3^2$ $3^{3x - 4x - 8y} = 3^2$ $3^{-x - 8y} = 3^2$

Since the bases are the same, we equate the exponents: $-x - 8y = 2$ --- (Equation 3)

Step 2: Solve the system of linear equations

We now have:

1. $x + 4y = 0$ $\implies x = -4y$
2. $-x - 8y = 2$

Substitute $x = -4y$ into Equation 3: $-(-4y) - 8y = 2$ $4y - 8y = 2$ $-4y = 2$ $y = \frac{2}{-4} = -\frac{1}{2}$

Now, find $x$ using $x = -4y$: $x = -4(-\frac{1}{2})$ $x = 2$

Step 3: Verify the results

• $x = 2, y = -1/2$
• Check Equation 2: $x + 4y = 2 + 4(-1/2) = 2 - 2 = 0$. (Correct)
• Check Equation 1: $3^{-x - 8y} = 3^{-(2) - 8(-1/2)} = 3^{-2 + 4} = 3^2 = 9$. (Correct)

The values are $x = 2$ and $y = -1/2$. This matches option B.

Exam Tip: Whenever you see equations involving exponents like 27, 81, 9, or 125, 25, 5, always try to express them as powers of a common base (usually the smallest prime number involved). This turns a complex exponential equation into a simple linear equation!