The equation of the straight line through the point `(-1,-2)` with slope `(4)/(7)` is............

Answer

To find the equation of the straight line that passes through the point \((-1, -2)\) with a slope of \(\frac{4}{7}\), we will use the point-slope form of the equation of a line. The point-slope form is given by:

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

where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

### Step-by-Step Solution:

1. **Identify the values**:
- The point given is \((-1, -2)\), so \(x_1 = -1\) and \(y_1 = -2\).
- The slope \(m\) is given as \(\frac{4}{7}\).

2. **Substitute the values into the point-slope form**:
\[ y - (-2) = \frac{4}{7}(x - (-1)) \]
This simplifies to:
\[ y + 2 = \frac{4}{7}(x + 1) \]

3. **Distribute the slope on the right side**:
\[ y + 2 = \frac{4}{7}x + \frac{4}{7} \]

4. **Isolate \(y\)**:
Subtract \(2\) from both sides:
\[ y = \frac{4}{7}x + \frac{4}{7} - 2 \]
To combine the constants, convert \(2\) to a fraction with a denominator of \(7\):
\[ 2 = \frac{14}{7} \]
Thus:
\[ y = \frac{4}{7}x + \frac{4}{7} - \frac{14}{7} \]
This simplifies to:
\[ y = \frac{4}{7}x - \frac{10}{7} \]

5. **Rearranging to standard form**:
To express this in standard form \(Ax + By + C = 0\), we can multiply through by \(7\) to eliminate the fraction:
\[ 7y = 4x - 10 \]
Rearranging gives:
\[ 4x - 7y - 10 = 0 \]

### Final Answer:
The equation of the straight line is:
\[ 4x - 7y - 10 = 0 \]