Final answer:
To find the equation of the bisector of the acute angle formed by two lines, we must determine the slopes of the given lines and then utilize those to find the slope of the angle bisector, finally using this information to write the bisector's equation in the point-slope form.
Step-by-step explanation:
The student is asking for the equation of the bisector of the acute angle formed by two given lines. To find this equation, we need to determine the slopes of the given lines, then find the slope of the angle bisector, which is the average of the slopes if the lines are perpendicular (which they are not in this instance). Once we have the slope of the angle bisector, we can use the point-slope form to find the equation. Details provided by Figure A1 regarding slope and the algebra of straight lines help us understand that for a line with equation y = mx + b, m represents the slope, while b represents the y-intercept.
For the given lines x+3y-6=0 and 3x+y+2=0, we would first rewrite them in the slope-intercept form to obtain their slopes. After finding the slopes, we would use the formulas to calculate the slope of the angle bisector. This calculation involves using geometric principles to establish the correct angle bisector (acute or obtuse) and would result in an equation that can be compared to the standard form y = mx + b. However, without the actual calculations and numerical values for the slopes of the given lines, we cannot provide a numerical answer.