Geometry equation help?

Question

What is the equation of the line through points
(-3, 2) and (-4, 5)?

What is the equation of the line through points
(2, 5) and (5, 7)?

What is the equation of the line through point (3, 2) and parallel to a line with slope 2?

Answer 1

The general equation for a straight line is
y = mx + c
Where 'm' in the gradient and 'c' is the intercept on the y-axis.
So for (-3,2) & (-4,5)
Gradient(m) is
(5 - 2)/(-4 --3)
3/-1
=-3
y = -3x + c
The taking either of the two points and substituting in for 'x' and 'y'.
2 = -3(-3) + c
2 = 6 + c
c = 2 -6 = -4
y = -3x - 4

So for (2,5) & (5,7)
Gradient(m) is
(7 - 5)/(5 - 2)
2/3
= 2/3 (0.6666666666....)
y = (2/3)x + c
The taking either of the two points and substituting in for 'x' and 'y'.
7 = 2/3(5) + c
7 =10/3 + c
c = 7 -10/3 = 11/3
y = (2/3)x -11/3
3y = 2x -11

If the line is parallel it has the same slope of 2.
y = 2x + c
Substituting in (3,2)
2 = 2(3) + c
2 = 6 + c
c = 2 -6 = -4
y = 2x - 4