# Thread: equation of straight line

1. ## equation of straight line

hey,
write down the equation of the straight line that:
a) passes through (0,-10) and is parrallel to the line with equation y=6x+3
b) passes through the point (0,-1) and is perpendicular to the line with equation 3x-2y+5=0

thanks

2. Hello,

Originally Posted by fresh_
hey,
write down the equation of the straight line that:
a) passes through (0,-10) and is parrallel to the line with equation y=6x+3
If two lines are parallel, then the slopes are equal.

The slope of y=6x+3 is 6.
Therefore, the equation of the line we want to find is y=6x+a, with a to determine.
Because it passes through (0,-10), we have -10=6*0+a. So a=-10.

Thus the equatio of the line is $y=6x-10$

b) passes through the point (0,-1) and is perpendicular to the line with equation 3x-2y+5=0
y=ax+b

It passes through the point 0,-1. Therefore, -1=a*0+b --> b=-1

ax-y-1=0.

A normal vector to it is $(a,-1)$.

A normal vector to 3x-2y+5=0 is $(3,-2)$.

Because the two lines are perpendicular, their normal vector are perpendicular too. In particular, the scalar product is 0.

$0=(a,-1) \dot (3,-2)=3a+2 \implies a=-\frac 23$

You have the equation of the line now;..

i hope i didn't make any mistake...

3. Originally Posted by Moo
Hello,

If two lines are parallel, then the slopes are equal.

The slope of y=6x+3 is 6.
Therefore, the equation of the line we want to find is y=6x+a, with a to determine.
Because it passes through (0,-10), we have -10=6*0+a. So a=-10.

Thus the equatio of the line is $y=6x-10$

y=ax+b

It passes through the point 0,-1. Therefore, -1=a*0+b --> b=-1

ax-y-1=0.

A normal vector to it is $(a,-1)$.

A normal vector to 3x-2y+5=0 is $(3,-2)$.

Because the two lines are perpendicular, their normal vector are perpendicular too. In particular, the scalar product is 0.

$0=(a,-1) \dot (3,-2)=3a+2 \implies a=-\frac 23$

You have the equation of the line now;..

i hope i didn't make any mistake...
so equation of line is y= - 2/3x + -1

is their any way to arrange the equation to 3y + 2x =-3 ?

4. @Moo: You're mistake-free.
@fresh_: Yes, of course.

Here I listed the steps for you:
Given: $y= -\frac{2x}{3} - 1$

Move 1 to the other side: $y + 1 = -\frac{2x}{3}$

Cross multiply: $3y + 3 = -2x$

Switch sides with 2x and 3: $3y + 2x = -3$

OR you can simply do this:
$\left(y + \frac{2x}{3} = - 1\right) \cdot 3$

$3y + 2x = -3$