To find the value of [tex]\(\frac{8^y}{4^x}\)[/tex] given the equation [tex]\(2x - 3y = 3\)[/tex], follow these steps:
1. Express [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
Given the equation:
[tex]\[
2x - 3y = 3,
\][/tex]
we solve for [tex]\(y\)[/tex]:
[tex]\[
2x - 3y = 3 \quad \Rightarrow \quad -3y = 3 - 2x \quad \Rightarrow \quad y = \frac{2x - 3}{3}.
\][/tex]
2. Substitute [tex]\(y\)[/tex] into [tex]\(\frac{8^y}{4^x}\)[/tex]:
We know:
[tex]\[
y = \frac{2x - 3}{3}.
\][/tex]
First, recall the properties of exponents:
[tex]\[
8 = 2^3 \quad \text{and} \quad 4 = 2^2.
\][/tex]
So,
[tex]\[
8^y = (2^3)^y = 2^{3y} \quad \text{and} \quad 4^x = (2^2)^x = 2^{2x}.
\][/tex]
Thus,
[tex]\[
\frac{8^y}{4^x} = \frac{2^{3y}}{2^{2x}} = 2^{3y - 2x}.
\][/tex]
3. Substitute [tex]\(y = \frac{2x - 3}{3}\)[/tex] into the exponent:
Substitute [tex]\(y\)[/tex] in [tex]\(2^{3y - 2x}\)[/tex]:
[tex]\[
3y = 3 \left(\frac{2x - 3}{3}\right) = 2x - 3.
\][/tex]
Therefore:
[tex]\[
3y - 2x = (2x - 3) - 2x = -3.
\][/tex]
Hence,
[tex]\[
2^{3y - 2x} = 2^{-3}.
\][/tex]
4. Calculate the value:
Since,
[tex]\[
2^{-3} = \frac{1}{2^3} = \frac{1}{8}.
\][/tex]
Thus, the value of [tex]\(\frac{8^y}{4^x}\)[/tex] given the equation [tex]\(2x - 3y = 3\)[/tex] is [tex]\(\boxed{\frac{1}{8}}\)[/tex].
