The first step to solve this problem would be to determine the slope of the line that passes through the two points. This slope, or gradient, can be calculated using the formula m = (y2 - y1) / (x2 - x1). In this case, point1 is (4, -13) and point2 is (-1, 7), so the slope would be computed as m = (7 - -13) / (-1 - 4) = (-4).
Therefore, the slope of the line passing through our points is -4.
We then can use one of our points and the slope we found to write the equation of the line in point-slope form: y - y1 = m(x - x1). If we use point1 for this, we get: y - -13 = -4(x - 4).
Simplifying the equation will get us to the 'slope-intercept form', which is the form y = mx + c, where m is the slope and c is the y-intercept. To find c, simplify the point-slope equation: y = -4x + 16 - 13, which simplifies to y = -4x + 3.
In conclusion, the equation of the line passing through points (4, -13) and (-1, 7) is y = -4x + 3.