Confusion about writing the standard forms of the equations of the lines that are parallel and perpendicular to y = 2x-3 and pass through the point-

[GetThroughDiffEq]

with the given coordinates.

I'm supposed to solve using analytic methods. That being said, my book doesn't offer an example.

12. (-3,6)

Work:

Slope of y=2x-3 is -3/2

y-y₁=m(x-x₁)

y-6=(-3/2)(x--3)

y-6==-(3/2x)-(9/2)
+6 +6

y=-3/2x+1.5

My answer: -1.5x-y+1.5

 

[pka]

with the given coordinates. I'm supposed to solve using analytic methods. That being said, my book doesn't offer an example.
12. (-3,6)
Work:
Slope of y=2x-3 is -3/2 Wrong slope is 2
y-y₁=m(x-x₁) CORRECT!

If \(\displaystyle \ell: ax+by+c=0\) is a line where \(\displaystyle a\cdot b\ne 0\) and \(\displaystyle (p,q)\) is a point
then the line \(\displaystyle ax+by-(ap+bq)=0\) is parallel to \(\displaystyle \ell\) through \(\displaystyle (p,q)\).
The line \(\displaystyle bx-ay-(bp-aq)=0\) is perpendicular to \(\displaystyle \ell\) through \(\displaystyle (p,q)\).

 

[JeffM]

with the given coordinates.

I'm supposed to solve using analytic methods. That being said, my book doesn't offer an example.

12. (-3,6)

Work:

Slope of y=2x-3 is -3/2

y-y₁=m(x-x₁)

y-6=(-3/2)(x--3)

y-6==-(3/2x)-(9/2)
+6 +6

y=-3/2x+1.5

My answer: -1.5x-y+1.5

[MATH]-\ 1.5x - y + 1.5[/MATH] is not even an equation.

The slope of y = 2x - 3 is 2 rather than -1.5.

Any line parallel to y = 2x - 3 will also have a slope of 2 and the slope-intercept form of y = 2x + z.

Any line perpendicular to 2x - 3 will have a slope of [MATH]-\ \dfrac{1}{2} = -\ 0.5[/MATH] and the slope intercept form of y = w - 0.5x.

Figure out what w and z are and express in standard form.

If you find these problems difficult, postpone taking differential equations.

 

[Steven G]

An equation of a line is a relation between x and y. How can you describe the relation between x and y without an equation? An equation has an equal sign!

 

[pka]

with the given coordinates. I'm supposed to solve using analytic methods. That being said, my book doesn't offer an example.
12. (-3,6)

Allow me to give you an example.
Given a line \(\displaystyle \ell: y=2x-3\) and a point \(\displaystyle P: (-3,6)\).
The standard form of the line \(\displaystyle \ell\) is \(\displaystyle 2x-y-3=0\)
Now any line parallel to \(\displaystyle \ell\) looks like \(\displaystyle 2x-y+d=0\)
If the line is to contain \(\displaystyle P\) then its coordinates must satisfy that equation.
That means \(\displaystyle d=-[2(-3)-(6)]\) OR \(\displaystyle d=12\).

Now we address the perpendicular case.
Any line perpendicular to \(\displaystyle \ell\) looks like \(\displaystyle x+2y+g=0\)
If that line contains \(\displaystyle P\) then what if the value of \(\displaystyle g~?\)