Question

Gillian swears her computations for the

following equations prove they do not
intersect. Her brother who just finished
learning about intersecting lines told her they
definitely intersect because the slopes are
different. Gillian remembered that logic from
class and then decided she needed to be able
to prove intersection by using algebra.
Although there are multiple strategies, how
might she prove intersection without graphing
of the following equations?
4x +3y = 6 and 6x + 2y = 10

Answer 1

Explanation:

Given

Two lines are
`4x+3y=6` and
`6x+2y=10`

Two lines
`a_1x+b_1y=c_1` and
`a_2x+b_2y=c_2` will intersect when

`(a_1)/(a_2)\\eq (b_1)/(b_2)`

for the given lines

`a_1=4,a_2=6,b_1=3,b_2=2`

`\therefore (4)/(6)\\eq (3)/(2)\\\\(2)/(3)\\eq(3)/(2)`

Hence, lines are intersecting