Final answer:
To find the equation of a line that is perpendicular to another line, we need to determine the slope of the given line and then find the negative reciprocal of that slope. The given line is 5x - 6y = 4, which can be rearranged into the standard form of a line, y = mx + b. The equation of the line passing through (3, 1) that is perpendicular to 5x - 6y = 4 is y = (-6/5)x + 23/5.
Step-by-step explanation:
To find the equation of a line that is perpendicular to another line, we need to determine the slope of the given line and then find the negative reciprocal of that slope. The given line is 5x - 6y = 4, which can be rearranged into the standard form of a line, y = mx + b, where m is the slope. So, let's rearrange the equation to get in the form y = mx + b:
5x - 6y = 4 => -6y = -5x + 4 => y = (5/6)x - 4/6 => y = (5/6)x - 2/3.
Since we want a line perpendicular to this line, the negative reciprocal of the slope, (5/6), will be the slope of the new line. The negative reciprocal of (5/6) is -6/5. Now, we can plug in the given point (3, 1) and the slope -6/5 into the point-slope form of the line, y - y1 = m(x - x1):
y - 1 = (-6/5)(x - 3)
Finally, let's rearrange this equation into slope-intercept form, y = mx + b:
y - 1 = (-6/5)(x - 3) => y - 1 = (-6/5)x + 18/5 => y = (-6/5)x + 18/5 + 5/5 => y = (-6/5)x + 23/5.