To find the equation of the line through the points \((2, 1)\) and \((5, -8)\), we first need to determine the slope \(m\) of the line.
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Assigning the points \((x_1, y_1) = (2, 1)\) and \((x_2, y_2) = (5, -8)\), we substitute these values into the formula: \[ m = \frac{-8 - 1}{5 - 2} = \frac{-9}{3} = -3 \]
Now that we have the slope, we can use the point-slope form of a linear equation. The point-slope form is expressed as: \[ y - y_1 = m(x - x_1) \]
Using the point \((2, 1)\) and the slope \(m = -3\): \[ y - 1 = -3(x - 2) \]
Next, we will simplify this equation: \[ y - 1 = -3x + 6 \] \[ y = -3x + 7 \]
Therefore, the equation of the line is: \[ \boxed{y = -3x + 7} \]
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