Given two points A(3, 12) and B(5, 22) on an \(x-y\) plane, we can find the slope of the line passing through these two points using the formula for slope:

\(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\)

Substituting the coordinates of points \(A\) and \(B\) into this formula, we get:

\(m = \frac{{22 - 12}}{{5 - 3}} = \frac{{10}}{{2}} = 5\)

So, the slope of the line passing through points A and B is 5.

The equation of a straight line can be written in slope-intercept form as \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. Since we know that the slope of the line is 5 and it has a y-intercept at 2, we can write the equation of the line as:

y = 5x + 2