2. Write the equation of the line through the two indicated points. Write the final answer in the form \( A x+B y=C \). (A) \( (-3,5),(1,-1) \) (B) \( (-1,5),(4,5) \)

To find the equation of the line through the points \((-3,5)\) and \((1,-1)\), we first calculate the slope \(m\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 5}{1 - (-3)} = \frac{-6}{4} = -\frac{3}{2} \] Using the point-slope form of the line equation \(y - y_1 = m(x - x_1)\) with point \((-3, 5)\): \[ y - 5 = -\frac{3}{2}(x + 3) \] Expanding and rearranging gives us: \[ y = -\frac{3}{2}x - \frac{9}{2} + 5 \] \[ y = -\frac{3}{2}x + \frac{1}{2} \] To convert to the desired form \(Ax + By = C\), multiply by 2: \[ 2y = -3x + 1 \implies 3x + 2y = 1 \] For points \((-1,5)\) and \((4,5)\), since the y-values are the same, the line is horizontal: The equation can be expressed as \(y = 5\). Putting it in \(Ax + By = C\) gives us: \[ 0x + 1y = 5 \implies 0x + 1y = 5 \] Therefore, the line equations in the desired format are: For points (-3, 5) and (1, -1): \(3x + 2y = 1\) For points (-1, 5) and (4, 5): \(0x + 1y = 5\)