\( x+2 y=6 \) Write an equation of the lime that passes through the given point is paralle1 to the graph of the givan equation. \( \begin{array}{ll}\text { 27) }(-1,3) ; y=2 x-8 & \text { 28.) }(-3,12): y=\frac{-1}{3} x+7\end{array} \)

To find the equation of the line that is parallel to \(x + 2y = 6\) and passes through the point \((-1, 3)\), we first need the slope of the given line. Rewriting the equation in slope-intercept form \(y = mx + b\): 1. Rearranging gives \(2y = -x + 6\) leading to \(y = -\frac{1}{2}x + 3\). The slope (\(m\)) is \(-\frac{1}{2}\). 2. Since parallel lines share the same slope, the new line will also have a slope of \(-\frac{1}{2}\). Using the point-slope form \(y - y_1 = m(x - x_1)\): \[ y - 3 = -\frac{1}{2}(x + 1) \] 3. Simplifying yields: \[ y - 3 = -\frac{1}{2}x - \frac{1}{2} \Rightarrow y = -\frac{1}{2}x + 2.5 \] So, the equation of the line is \(y = -\frac{1}{2}x + 2.5\). Now, for the second point \((-3, 12)\), we first find the slope of the line given as \(y = \frac{-1}{3}x + 7\), which has a slope of \(-\frac{1}{3}\). 1. Using the same process with the point-slope form: \[ y - 12 = -\frac{1}{3}(x + 3) \] 2. Simplifying gives: \[ y - 12 = -\frac{1}{3}x - 1 \Rightarrow y = -\frac{1}{3}x + 11 \] So, the equation of the line through \((-3, 12)\) is \(y = -\frac{1}{3}x + 11\).