Find the equation of a straight line passing through (−1,−2) having slope `1/3`

Explanation

To find the equation of a straight line passing through the point \((-1, -2)\) with a slope of \(\frac{1}{3}\), we can use the point-slope form of the equation of a line, which is given by:

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

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

### Step 1: Identify the given values
We have:
- Point: \((x_1, y_1) = (-1, -2)\)
- Slope: \(m = \frac{1}{3}\)

### Step 2: Substitute the values into the point-slope formula
Substituting the values into the formula:

\[ y - (-2) = \frac{1}{3}(x - (-1)) \]

This simplifies to:

\[ y + 2 = \frac{1}{3}(x + 1) \]

### Step 3: Distribute the slope on the right side
Now, we will distribute \(\frac{1}{3}\) on the right side:

\[ y + 2 = \frac{1}{3}x + \frac{1}{3} \]

### Step 4: Isolate \(y\)
Next, we will isolate \(y\) by subtracting \(2\) from both sides:

\[ y = \frac{1}{3}x + \frac{1}{3} - 2 \]

To combine the constants, we convert \(2\) into a fraction with a denominator of \(3\):

\[ y = \frac{1}{3}x + \frac{1}{3} - \frac{6}{3} \]

This simplifies to:

\[ y = \frac{1}{3}x - \frac{5}{3} \]

### Step 5: Rearranging to standard form
To express the equation in standard form \(Ax + By + C = 0\), we can rearrange it:

\[ -\frac{1}{3}x + y + \frac{5}{3} = 0 \]

Multiplying through by \(3\) to eliminate the fraction gives:

\[ -x + 3y + 5 = 0 \]

Rearranging this, we get:

\[ x - 3y = 5 \]

### Final Equation
Thus, the equation of the straight line is:

\[ x - 3y = 5 \]