Final answer:
The equation of the line perpendicular to -2x + 5y = 6 with an x-intercept at 3 is y = -5/2x + 15/2, aligning with option d). This is found by calculating the negative reciprocal of the original line's slope and using the point-slope form of a linear equation.
Step-by-step explanation:
To find an equation in slope-intercept form for a line that is perpendicular to another and has a given x-intercept, we need to determine two things: the slope of the line we're seeking (which must be the negative reciprocal of the slope of the given line), and its y-intercept.
Firstly, let's find the slope of the given line -2x + 5y = 6.
We can rewrite it in slope-intercept form (y = mx + b).
Doing so, we get y = (2/5)x + 6/5. Therefore, the slope (m) of the given line is 2/5.
The slope of a line perpendicular to this would be the negative reciprocal, which is -5/2.
Next, since the line has an x-intercept at 3, the coordinates of the x-intercept are (3, 0).
We now use the slope we found and this point to write the equation of our new line.
Starting with the slope-intercept form y = mx + b and substituting m = -5/2 and the coordinates of the point, we have 0 = (-5/2)(3) + b. Solving for b gives us b = 15/2.
Therefore, the equation of the line we're looking for is y = -5/2x + 15/2, which matches option d).