1 Determine the linear equation for the line that goes through the following coordinates: \( \begin{array}{l}\text { Q.9.1.1 } \\ \text { Q.5.10) and }(-3,4) \\ (-4,-22) \text { and }(-6,-34)\end{array} \)

To find the linear equation for the line that goes through the points (-3, 4) and (-4, -22), we first need to calculate the slope (m) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using the coordinates, we have \( m = \frac{-22 - 4}{-4 - (-3)} = \frac{-26}{-1} = 26 \). Now, use the point-slope form of a linear equation \( y - y_1 = m(x - x_1) \) with one of the points, say (-3, 4). Plugging in, we get \( y - 4 = 26(x + 3) \). Simplifying yields the equation \( y = 26x + 82 \). Now, let's try the coordinates (-6, -34) and see if they fit this equation. Plugging in, we find \( -34 = 26(-6) + 82 \), which simplifies to \( -34 = -156 + 82 \) or \( -34 = -74, \) a contradiction indicating that not all points are on the same line. Thus, the unique equation derived from the first two points is \( y = 26x + 82 \).