Which BEST describes the relationship between the lines with equations \( -4 x+2 y=-5 \) and \( -8 x+4 y=0 \) ? A. neither parallel nor perpendicular B. perpendicular C. parallel D. same line

To determine the relationship between the two lines given by the equations: 1. \( -4x + 2y = -5 \) 2. \( -8x + 4y = 0 \) **Step 1: Find the slopes of both lines.** First, rewrite both equations in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. **For the first line:** \[ -4x + 2y = -5 \\ 2y = 4x - 5 \\ y = 2x - \frac{5}{2} \] So, the slope \( m_1 = 2 \). **For the second line:** \[ -8x + 4y = 0 \\ 4y = 8x \\ y = 2x \] So, the slope \( m_2 = 2 \). **Step 2: Compare the slopes.** Both lines have the same slope (\( m_1 = m_2 = 2 \)), which means they are either **parallel** or **the same line**. **Step 3: Check if they are the same line.** To determine if they are the same line, see if one equation is a multiple of the other. Multiplying the first equation by 2: \[ 2(-4x + 2y) = 2(-5) \\ -8x + 4y = -10 \] This is not the same as the second equation (\( -8x + 4y = 0 \)), so they are **not** the same line. **Conclusion:** Since the lines have the same slope but are not the same line, they are **parallel**. **Answer:** C. parallel