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To convert the Cartesian equation \( y = 2x - 1 \) into polar coordinates, we need to use the relationships between Cartesian and polar coordinates:

\[
x = r \cos \theta
\]
\[
y = r \sin \theta
\]

### Step 1: Express \( y = 2x - 1 \) in terms of \( x \) and \( y \)

The equation is already in Cartesian form, so we have:

\[
y = 2x - 1
\]

### Step 2: Substitute the polar coordinate relationships

Replace \( y \) with \( r \sin \theta \) and \( x \) with \( r \cos \theta \):

\[
r \sin \theta = 2r \cos \theta - 1
\]

### Step 3: Solve for \( r \)

Rearranging the terms:

\[
r \sin \theta - 2r \cos \theta = -1
\]

Factor out \( r \):

\[
r (\sin \theta - 2 \cos \theta) = -1
\]

Now solve for \( r \):

\[
r = \frac{-1}{\sin \theta - 2 \cos \theta}
\]

Thus, the polar form of the equation \( y = 2x - 1 \) is:

\[
r = \frac{-1}{\sin \theta - 2 \cos \theta}
\]

---

Would you like more details about this transformation or any specific part of the process? 

Here are 5 related questions that can deepen your understanding:

1. How do you convert other linear equations from Cartesian to polar form?
2. What is the geometrical interpretation of the polar form of a line?
3. Can you convert nonlinear Cartesian equations into polar form?
4. How would this equation change if the slope in the original Cartesian equation was different?
5. What happens to the polar form if the constant term in the Cartesian equation changes?

**Tip:** When converting Cartesian equations into polar form, always remember that \( r \) represents the distance from the origin to a point, and \( \theta \) is the angle from the positive \( x \)-axis.

Write the following Cartesian curve in polar form: y = 2x - 1

Learn how to convert the linear Cartesian equation y = 2x - 1 into polar form using fundamental polar coordinate equations like x = r * cos(θ) and y = r * sin(θ). This step-by-step guide simplifies the conversion process, ideal for students in Grades 10-12.

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