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. **Find the Slope ($$ m $$) of the Line:**

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

2. **Use the Point-Slope Form of the Equation:**

The point-slope form of a line passing through a point $$ (x_1, y_1) $$ with slope $$ m $$ is:
   $$
   y - y_1 = m(x - x_1)
   $$
   
   Substituting $$ m = -1 $$ and $$ (x_1, y_1) = (3, 1) $$:
   $$
   y - 1 = -1(x - 3)
   $$
   
   Simplifying:
   $$
   y - 1 = -x + 3
   $$
   $$
   y = -x + 4
   $$
   
   This can also be written in standard form as:
   $$
   x + y = 4
   $$

**Final Answer:**
$$
y = -\,x + 4
$$
Alternatively, in standard form:
$$
x + y = 4
$$
The equation of the line is $$ y = -x + 4 $$ or $$ x + y = 4 $$.

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

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