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

To find the equation of the line, we first determine the slope \( m \) using the angle given. The slope can be calculated as \( m = \tan(\theta) \). For an angle of \( 135^{\circ} \), the slope is \( m = \tan(135^{\circ}) = -1 \). Now, using the point-slope form of the line equation \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point \( (3, 1) \), we can substitute our values: \[ y - 1 = -1(x - 3) \] This simplifies to: \[ y - 1 = -x + 3 \implies y = -x + 4 \] So, the equation of the line is \( y = -x + 4 \). To visualize the impact of this line, think about how lines with different slopes behave. A slope of -1 indicates a steadily descending line; here's an eye-opener for art enthusiasts: artists often use lines with such slopes to create dramatic contrasts in their compositions, highlighting emotional tension. It goes to show that math and art are more related than you might think! Next, consider the importance of angles in real-world scenarios. For example, architects might use lines at specific angles to optimize the aesthetics and functionality of a building's design. A line at \( 135^{\circ} \) can create dynamic, visually engaging spaces, while also letting in natural light from specific angles. So next time you see a building, think about how those angles play a role beyond just structure!