Find the equation of the line passing through \( (7,-2) \) and parallel to \( 2 x+7 y=21 \).

To find the equation of a line parallel to a given line, we need to determine the slope of the given line and then use that slope to find the equation of the parallel line passing through a given point. Given line: \(2x + 7y = 21\) To find the slope of the given line, we need to rewrite it in slope-intercept form \(y = mx + b\), where \(m\) is the slope. 1. Rewrite the given line in slope-intercept form: \[2x + 7y = 21\] \[7y = -2x + 21\] \[y = -\frac{2}{7}x + 3\] The slope of the given line is \(-\frac{2}{7}\). Since the line we are looking for is parallel to the given line, it will have the same slope of \(-\frac{2}{7}\). Now, we have the slope of the parallel line and a point it passes through \((7, -2)\). We can use the point-slope form of the equation of a line to find the equation of the parallel line. The point-slope form of the equation of a line is: \[y - y_1 = m(x - x_1)\] where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Substitute the values of the point \((7, -2)\) and the slope \(-\frac{2}{7}\) into the point-slope form: \[y - (-2) = -\frac{2}{7}(x - 7)\] \[y + 2 = -\frac{2}{7}x + 2\] \[y = -\frac{2}{7}x\] Therefore, the equation of the line passing through \((7, -2)\) and parallel to \(2x + 7y = 21\) is \(y = -\frac{2}{7}x\).