Question

Find the equations of the following lines

b. Gradient -4 and passing through point (2, 1)
c. Passing through the points (2, -1) and (4, 2)
d. Passing through the points (1, -3) and (6, -5)
e. Passing through the point (5, -2) and parallel to x + 5y + 15 = 0
f. Passing through the point (1, 6) and parallel to x - 3y - 2 = 0
g. Passing through the point (-1, -5) and perpendicular to 3x + y + 2 = 0

Answer 1

You need these two basic solutions and facts to find the equation of a line:

• If you know the gradient
  `m` and one point
  `(x_0,y_0)`:

`y-y_0=m(x-x_0)`

• If you know the gradient two points
  `(x_1,y_1),\ (x_2,y_2)`:

`(x-x_2)/(x_1-x_2)=(y-y_2)/(y_1-y_2)`

• The slope of a line is the coefficient m when you write it in the
  `y=mx+q` form
• Parallel lines have the same slope
• The slopes of perpendicular lines give -1 when multiplied

We can use this list to solve all the exercises:

b)

Use the first equation to get

`y-1=-4(x-2) \iff y=-4x+9`

c)

Use the second equation to get

`(x-4)/(2-4)=(y-2)/(-1-2) \iff (x-4)/(-2)=(y-2)/(-3)\iff 3(x-4)=2(y-2) \iff 3x-12=2y-4 \iff 2y = 3x-8 \iff y = (3)/(2)x-4`

d) same as c)

e) We derive the slope of the line by writing it as

`5y = -x-15 \iff y = -(1)/(5)x-3`

So, the slope is -1/5. From here, it's the same as b)

f) same as e)

g) Again we find the slope as

`3x+y+2=0\iff y=-3x-2`

so the slope is -3, and a perpendicular line has slope 1/3. From there, it's the same as b).