If two lines are parallel to each other, they both have the same slope. So let us find the slope we are looking for:
5x + 2y = 12
2y = -5x + 12
y = -5/2x + 6
We want a line with slope -5/2, and it has to pass through (-10,3).
We can plug in this given point to solve:
3 = -5/2(-10) + ?
3 = 25 + ?
28 is our answer for that last question and so it is our y-intercept.
So the equation of our first line is y = -5/2x - 22
If two lines are perpendicular, their slopes are negative reciprocals of each other. So first find the slope of the first line:
5x + 3y = 15
3y = -5x + 15
y = -5/3x + 5
Your slope is -5/3 and the negative reciprocal of that is 3/5.
So we are looking for a line with slope 3/5 that goes through point (5,1).
Lets plug this point in, just like last time:
y = mx + b
1 = 3/5(5) + b
1 = 3 + b
b = -2
So the equation of this line is y = 3/5x - 2.
Hope this helps! :)