Find the equations to the straight lines passing through the following pairs of points.
`(-1, 3) and (6, -7)`

Answer

To find the equation of the straight line passing through the points \((-1, 3)\) and \((6, -7)\), we can follow these steps:

### Step 1: Identify the points
Let the points be:
- Point A: \((x_1, y_1) = (-1, 3)\)
- Point B: \((x_2, y_2) = (6, -7)\)

### Step 2: Use the point-slope form of the line
The equation of the line in point-slope form is given by:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]

### Step 3: Substitute the values into the formula
Substituting the coordinates of points A and B into the formula:
\[ y - 3 = \frac{-7 - 3}{6 - (-1)} (x - (-1)) \]

### Step 4: Simplify the equation
Calculate the slope:
\[ \frac{-7 - 3}{6 - (-1)} = \frac{-10}{6 + 1} = \frac{-10}{7} \]
Now substitute back into the equation:
\[ y - 3 = \frac{-10}{7} (x + 1) \]

### Step 5: Distribute and rearrange
Distributing the slope:
\[ y - 3 = \frac{-10}{7}x - \frac{10}{7} \]
Now, add 3 (which is \(\frac{21}{7}\)) to both sides:
\[ y = \frac{-10}{7}x - \frac{10}{7} + \frac{21}{7} \]
Combine the constants:
\[ y = \frac{-10}{7}x + \frac{11}{7} \]

### Step 6: Convert to standard form
To convert this to standard form \(Ax + By + C = 0\):
Multiply through by 7 to eliminate the fraction:
\[ 7y = -10x + 11 \]
Rearranging gives:
\[ 10x + 7y - 11 = 0 \]

### Final Answer
The equation of the straight line passing through the points \((-1, 3)\) and \((6, -7)\) is:
\[ 10x + 7y - 11 = 0 \]