Final Answer:
The equation of the lines perpendicular to the line 6y - 12x = 7 is 2x + y = C, where C is a constant.
Step-by-step explanation:
To determine the equation of lines perpendicular to the given line 6y - 12x = 7, it's essential to understand that perpendicular lines in a Cartesian plane have slopes that are negative reciprocals of each other. The equation of the given line, 6y - 12x = 7, can be rewritten in slope-intercept form (y = mx + b), where m represents the slope. Rearranging the equation, we get y = 2x + 7/6, which indicates that the slope of this line is 2.
For lines perpendicular to this, we need to find a slope that is the negative reciprocal of 2. The negative reciprocal of 2 is -1/2. Therefore, any line perpendicular to 6y - 12x = 7 will have a slope of -1/2.
The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Substituting the perpendicular slope (-1/2) into the equation, we get y = (-1/2)x + b. The value of b (the y-intercept) can vary, so it can be represented as y = (-1/2)x + C, where C is any constant. Therefore, the equation of lines perpendicular to 6y - 12x = 7 is 2x + y = C, where C represents any constant value.