b) The line perpendicular to \( y=-x+3 \) c) The line parallel to \( y=-\frac{1}{2} x-1 \) d) The line perpendicular to \( y=\frac{2}{3} x-4 \) et The line perpendicular to \( y=-\frac{1}{4} x+1 \)

To find the equations of lines that are either perpendicular or parallel to the given lines, we need to follow these steps: 1. **Identify the slope of the given line.** 2. **For parallel lines:** The slope remains the same. 3. **For perpendicular lines:** The slope is the negative reciprocal of the given line's slope. Let's solve each part step by step. ### b) The line perpendicular to \( y = -x + 3 \) 1. The slope of the given line \( y = -x + 3 \) is \( m = -1 \). 2. The slope of the perpendicular line is the negative reciprocal: \[ m_{\text{perpendicular}} = -\frac{1}{-1} = 1 \] 3. The equation of the line can be written in point-slope form as \( y - y_1 = m(x - x_1) \). We can choose any point, but let's use the y-intercept of the original line, which is \( (0, 3) \): \[ y - 3 = 1(x - 0) \implies y = x + 3 \] ### c) The line parallel to \( y = -\frac{1}{2} x - 1 \) 1. The slope of the given line \( y = -\frac{1}{2} x - 1 \) is \( m = -\frac{1}{2} \). 2. The slope of the parallel line remains the same: \[ m_{\text{parallel}} = -\frac{1}{2} \] 3. Using the same point as before \( (0, -1) \): \[ y + 1 = -\frac{1}{2}(x - 0) \implies y = -\frac{1}{2}x - 1 \] ### d) The line perpendicular to \( y = \frac{2}{3} x - 4 \) 1. The slope of the given line \( y = \frac{2}{3} x - 4 \) is \( m = \frac{2}{3} \). 2. The slope of the perpendicular line is: \[ m_{\text{perpendicular}} = -\frac{1}{\frac{2}{3}} = -\frac{3}{2} \] 3. Using the y-intercept \( (0, -4) \): \[ y + 4 = -\frac{3}{2}(x - 0) \implies y = -\frac{3}{2}x - 4 \] ### e) The line perpendicular to \( y = -\frac{1}{4} x + 1 \) 1. The slope of the given line \( y = -\frac{1}{4} x + 1 \) is \( m = -\frac{1}{4} \). 2. The slope of the perpendicular line is: \[ m_{\text{perpendicular}} = -\frac{1}{-\frac{1}{4}} = 4 \] 3. Using the y-intercept \( (0, 1) \): \[ y - 1 = 4(x - 0) \implies y = 4x + 1 \] ### Summary of Results - b) The line perpendicular to \( y = -x + 3 \) is \( y = x + 3 \). - c) The line parallel to \( y = -\frac{1}{2} x - 1 \) is \( y = -\frac{1}{2} x - 1 \). - d) The line perpendicular to \( y = \frac{2}{3} x - 4 \) is \( y = -\frac{3}{2} x - 4 \). - e) The line perpendicular to \( y = -\frac{1}{4} x + 1 \) is \( y = 4x + 1 \).