Find the equation of the line which passes through the point (4, -5)and is
parallel to the line joining the points A(3,7) and B(-2,4) .

Answer

To find the equation of the line that passes through the point (4, -5) and is parallel to the line joining the points A(3, 7) and B(-2, 4), we can follow these steps:

### Step 1: Find the slope of the line joining points A and B.
The slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
\[ m = \frac{y2 - y1}{x2 - x1} \]

For points A(3, 7) and B(-2, 4):
- Let (x1, y1) = (3, 7) and (x2, y2) = (-2, 4).
- Substitute the values into the slope formula:
\[ m_{AB} = \frac{4 - 7}{-2 - 3} = \frac{-3}{-5} = \frac{3}{5} \]

### Step 2: Use the slope to find the equation of the parallel line.
Since the required line is parallel to line AB, it will have the same slope:
\[ m = \frac{3}{5} \]

### Step 3: Use the point-slope form of the equation of a line.
The point-slope form of the equation of a line is given by:
\[ y - y1 = m(x - x1) \]
Where (x1, y1) is the point through which the line passes.

Here, (x1, y1) = (4, -5) and m = \(\frac{3}{5}\):
\[ y - (-5) = \frac{3}{5}(x - 4) \]
This simplifies to:
\[ y + 5 = \frac{3}{5}(x - 4) \]

### Step 4: Simplify the equation.
Now, we will distribute the \(\frac{3}{5}\):
\[ y + 5 = \frac{3}{5}x - \frac{12}{5} \]

Next, we will move all terms to one side to write it in standard form:
\[ y = \frac{3}{5}x - \frac{12}{5} - 5 \]
Convert -5 to a fraction with a denominator of 5:
\[ -5 = -\frac{25}{5} \]
So,
\[ y = \frac{3}{5}x - \frac{12}{5} - \frac{25}{5} \]
\[ y = \frac{3}{5}x - \frac{37}{5} \]

### Step 5: Rearrange to standard form.
To rearrange this to standard form \(Ax + By + C = 0\):
Multiply through by 5 to eliminate the fraction:
\[ 5y = 3x - 37 \]
Rearranging gives:
\[ 3x - 5y - 37 = 0 \]

### Final Answer:
The equation of the required line is:
\[ 3x - 5y - 37 = 0 \]

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