through: \( (-1,5) \), parallel to \( y=-5 x+5 \)

To find the equation of a line that passes through the point \((-1, 5)\) and is parallel to the line \( y = -5x + 5 \), follow these steps: ### 1. **Determine the Slope of the Given Line** The given line is in **slope-intercept form**, which is: \[ y = mx + b \] where: - \( m \) is the slope - \( b \) is the y-intercept For the line \( y = -5x + 5 \): - **Slope (\( m \))** = \(-5\) ### 2. **Use the Slope for the Parallel Line** Parallel lines have **identical slopes**. Therefore, the slope of the line we want to find is also \(-5\). ### 3. **Apply the Point-Slope Formula** The **point-slope form** of a line equation is: \[ y - y_1 = m(x - x_1) \] where: - \( (x_1, y_1) \) is a point on the line - \( m \) is the slope Given the point \((-1, 5)\) and slope \( m = -5 \): \[ y - 5 = -5(x - (-1)) \] \[ y - 5 = -5(x + 1) \] ### 4. **Simplify to Slope-Intercept Form (Optional)** To express the equation in **slope-intercept form** (\( y = mx + b \)): \[ y - 5 = -5x - 5 \] \[ y = -5x - 5 + 5 \] \[ y = -5x \] ### **Final Equation** The equation of the line that passes through \((-1, 5)\) and is parallel to \( y = -5x + 5 \) is: \[ y = -5x \] ### **Graphical Representation** To visualize: 1. **Original Line:** \( y = -5x + 5 \) 2. **Parallel Line:** \( y = -5x \) 3. **Point on Parallel Line:** \((-1, 5)\) Both lines will have the same slope, meaning they descend at the same rate but are shifted vertically relative to each other.