Question

How do you find a standard form equation for the line with (5,4) perpendicular to the line 3x+2y=7?

Answer 1

`2x-3y=-2`

Explanation:

`3x+2y=7` has a slope of `-3/2color(white)("xxxx")` see Note 1

All lines perpendicular to `3x+2y=7` have a slope of `2/3color(white)("xxxx")`see Note 2

If such a perpendicular line goes through `(5,4)`
then we can write it equation in slope point form as:
`y-4=2/3(x-5)color(white)("xxxxxxxxxxxxx")`see Note 3

This can be converted into standard form as:
`2x-3y=-2color(white)("xxxxxxxxxxxxxxxx"`see Note 4

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Note 1
A relation in the form `Ax+By=C` has a slope of `-A/B`;
in this case `A=3` and `B=2`

If you are not familiar with this rule, you can convert given relation `3x+2y=7` into slope-intercept form:

`2y=-3x+7`

`y=(-3/2)x+7/2` with slope `(-3/2)` and y-intercept `7/2`

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Note 2
If a line has a slope of `m` then all line perpendicular to it have a slope of `(-1/m)`

In this case `-1/(-3/2)=2/3`

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Note 3

A line with slope `m` through a point `(x_0,y_0)`
has a slope-point form:
`color(white)("XXX")y-y_0=m(x-x_0)`

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Note 4
"standard form" for a linear equation is
`color(white)("XXX")Ax+By=C` with `A, B, C in ZZ, A>=0`

converting `y-4=2/3(x-5)` into this form:
`color(white)("XXX")3(y-4)=2(x-5)`

`color(white)("XXX")3y-12=2x-10`

`color(white)("XXX")-2x+3y=2`

`color(white)("XXX")2x-3y=-2`

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