Find the equation of a line passing through the point `(-1,-2)` and parallel to the line joining the points `(2,-3)` and `(3,-2)`

Answer

To find the equation of a line passing through the point \((-1, -2)\) and parallel to the line joining the points \((2, -3)\) and \((3, -2)\), we will follow these steps:

### Step 1: Find the slope of the line joining the points \((2, -3)\) and \((3, -2)\)

The formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, let \((x_1, y_1) = (2, -3)\) and \((x_2, y_2) = (3, -2)\).

Substituting the values:

\[ m = \frac{-2 - (-3)}{3 - 2} = \frac{-2 + 3}{1} = \frac{1}{1} = 1 \]

### Step 2: Use the point-slope form of the equation of a line

The point-slope form of the equation of a line is given by:

\[ y - y_1 = m(x - x_1) \]

We have the slope \(m = 1\) and the point \((-1, -2)\) which gives us \(x_1 = -1\) and \(y_1 = -2\).

Substituting these values into the point-slope form:

\[ y - (-2) = 1(x - (-1)) \]

This simplifies to:

\[ y + 2 = 1(x + 1) \]

### Step 3: Simplify the equation

Now, we simplify the equation:

\[ y + 2 = x + 1 \]

Subtracting 2 from both sides gives:

\[ y = x + 1 - 2 \]

Thus, we have:

\[ y = x - 1 \]

### Step 4: Rearranging to standard form

To express the equation in standard form \(Ax + By + C = 0\), we rearrange:

\[ x - y - 1 = 0 \]

This can also be written as:

\[ x - y = 1 \]

### Final Answer

The equation of the line passing through the point \((-1, -2)\) and parallel to the line joining the points \((2, -3)\) and \((3, -2)\) is:

\[ x - y = 1 \]