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 $(x_1,y_1),\ (x_2,y_2)$:

$\dfrac{x-x_2}{x_1-x_2}=\dfrac{y-y_2}{y_1-y_2}$

• 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

$\dfrac{x-4}{2-4}=\dfrac{y-2}{-1-2} \iff \dfrac{x-4}{-2}=\dfrac{y-2}{-3}\iff 3(x-4)=2(y-2) \iff 3x-12=2y-4 \iff 2y = 3x-8 \iff y = \frac{3}{2}x-4$

d) same as c)

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

$5y = -x-15 \iff y = -\dfrac{1}{5}x-3$

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).