4. Determine the equation of the straight line passing through the point $$ (3 ; 1) $$ and with an angle of $$ 135^{\circ} $$.

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To determine the equation of the straight line passing through the point $$ (3, 1) $$ and making an angle of $$ 135^\circ $$ with the positive $$ x $$-axis, follow these steps:

1. **Determine the Slope (m):**
   
   The slope of a line that makes an angle $$ \theta $$ with the positive $$ x $$-axis is given by:
   $$
   m = \tan(\theta)
   $$
   
   Given $$ \theta = 135^\circ $$:
   $$
   \tan(135^\circ) = \tan(180^\circ - 45^\circ) = -\tan(45^\circ) = -1
   $$
   
   So, the slope $$ m = -1 $$.

2. **Use the Point-Slope Form:**
   
   The point-slope form of a line is:
   $$
   y - y_1 = m(x - x_1)
   $$
   
   Plugging in the point $$ (3, 1) $$ and $$ m = -1 $$:
   $$
   y - 1 = -1(x - 3)
   $$
   $$
   y - 1 = -x + 3
   $$
   $$
   y = -x + 4
   $$

3. **Final Equation:**
   
   Thus, the equation of the line is:
   $$
   y = -x + 4
   $$

**Answer:** After simplifying, an equation for the line is y = – x + 4
The equation of the line is $$ y = -x + 4 $$.

Determine the equation of the straight line passing
through the point and with an angle of .

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