through: \( (1,-1) \), parallel to \( y=x-4 \)

To find the equation of a line that passes through the point \((1, -1)\) and is parallel to the line \(y = x - 4\), follow these steps: 1. **Determine the Slope of the Given Line:** The given line is \(y = x - 4\). This is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. - **Slope (\(m\))** of the given line: \(1\) 2. **Use the Point-Slope Form for the New Line:** Since parallel lines have the same slope, the new line will also have a slope of \(1\). The **point-slope form** is: \[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Plugging in the values: \[ y - (-1) = 1(x - 1) \] Simplify: \[ y + 1 = x - 1 \] \[ y = x - 2 \] 3. **Final Equation:** The equation of the line parallel to \(y = x - 4\) and passing through \((1, -1)\) is: \[ y = x - 2 \] **Graphical Representation:** - **Original Line:** \(y = x - 4\) (Slope = 1) - **New Line:** \(y = x - 2\) (Slope = 1, passes through (1, -1)) Both lines have the same slope, confirming they are parallel.