Can anyone please tell me how to derive the joint equation of 2 straight lines

suppose the 2 equations of 2 strt lines are: 2x+3y+9=0, and 5x-4y-7=0, then find the joint equation of those two lines. my teacher told me it is in the form of: ax^2+ 2hxy+ by^2+ 2gx+2fy+c=0, but i cant understand how it is derived...please explain..

Apparently the teacher did not make it clear that the joint equation of two straight lines is simply the product of their respective equations.

Thus the joint equation of ax+by+c = 0 and Ax+By+C = 0 is (ax+by+c)(Ax+By+C) = 0

For the two equations given, 2x+3y+9 = 0 and 5x-4y-7 = 0, the joint equation is

10x^2 + 7xy -12y^2 + 31x - 57y - 63, following the form given by your teacher
(I did this in my head, so please check the multiplication!)

The interesting thing about the joint equation is that it is true for any pair of values for which either of original equations is true.

For example 2x + 3y + 9 is true for -3, -1. So also is the joint equation. Try it and see.

1) Joint equation of two different straight lines is given by the product of the respective equtions.

2) As such here the product is: (2x + 3y + 9)*(5x - 4y - 7) = 0

==> 10(x^2) + 7xy - 12(y^2) + 31x - 57y - 63 = 0

So the joint equation is: 10(x^2) + 7xy - 12(y^2) + 31x - 57y - 63 = 0

If

u ≡ 2x + 3y + 9, v ≡ 5x - 4y - 7,

then, the joint equation of the two

lines u = 0 and v = 0 is :

u·v = 0,

that is,

( 2x + 3y + 9 )·( 5x - 4y - 7 ) = 0,

that is,

10x² + 7xy - 12y² - 31x - 57y - 63 = 0 ........... Ans.
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Happy To Help !
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