Find the equation of t he straight line through the given point P and having the given slope m if
`P(-4, 7), m=-sqrt(3)`

Answer

To find the equation of the straight line through the point \( P(-4, 7) \) with a slope \( m = -\sqrt{3} \), we can use the point-slope form of the equation of a line, which is given by:

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

### Step-by-Step Solution:

**Step 1: Identify the point and slope.**
- The point \( P \) is given as \( (-4, 7) \), so \( x_1 = -4 \) and \( y_1 = 7 \).
- The slope \( m \) is given as \( -\sqrt{3} \).

**Step 2: Substitute the values into the point-slope form.**
Using the point-slope form:

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

Substituting \( y_1 = 7 \), \( m = -\sqrt{3} \), and \( x_1 = -4 \):

\[ y - 7 = -\sqrt{3}(x - (-4)) \]

This simplifies to:

\[ y - 7 = -\sqrt{3}(x + 4) \]

**Step 3: Distribute the slope on the right-hand side.**
Distributing \( -\sqrt{3} \):

\[ y - 7 = -\sqrt{3}x - 4\sqrt{3} \]

**Step 4: Rearrange the equation.**
To write the equation in standard form, we can move all terms to one side:

\[ \sqrt{3}x + y - 7 + 4\sqrt{3} = 0 \]

This can be rearranged to:

\[ \sqrt{3}x + y + (4\sqrt{3} - 7) = 0 \]

**Final Equation:**
The equation of the straight line through the point \( P(-4, 7) \) with slope \( m = -\sqrt{3} \) is:

\[ \sqrt{3}x + y + (4\sqrt{3} - 7) = 0 \]