To convert the Cartesian equation 2x + 3y = 6 into polar coordinates, we first recall the relationships between Cartesian coordinates (x, y) and polar coordinates (r, θ):
x = r * cos(θ)
y = r * sin(θ)
1. **Substituting for x and y:**
Using these relationships, we can substitute for x and y in the given equation:
2(r * cos(θ)) + 3(r * sin(θ)) = 6
2r * cos(θ) + 3r * sin(θ) = 6
2r cos(θ) + 3r sin(θ) = 6
2r cos(θ) + 3r sin(θ) can be factored as:
r(2 cos(θ) + 3 sin(θ)) = 6
2. **Isolating r:**
Now, to isolate r, we divide both sides of the equation by (2 cos(θ) + 3 sin(θ)):
r = \( \frac{6}{2 cos(θ) + 3 sin(θ)} \
3. **Final Polar Equation:**
Thus, the polar equation that represents the curve described by the Cartesian equation 2x + 3y = 6 is:
r = \( \frac{6}{2 cos(θ) + 3 sin(θ)} \
This polar equation can now be used to further analyze the curve or to graph it in polar coordinates.