# What is the distance of the point where 2x + 3y = 9 and 6x + y = 4 intersect from (0, 2)

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the point of intersection between the lines, hence you need to solve simultaneous equations for x and y.

I suggest you to substitute `4 - 6x ` for y in the equation `2x+3y=9`  (notice that you may use the equation `6x + y = 4`  to write y in term of x).

Hence, you need to solve `2x + 3(4 - 6x) = 9` , such that:

`2x + 12 - 18x = 9 =gt -16x = 9 - 12 =gt -16x = -3 =gt x = 3/16`

Substsituting `3/16`  for x in `y= 4 - 6x`  yields:

`y = 4 - 6*3/16 =gt y = 4 - 9/8 =gt y = 23/8`

Hence, the lines `2x + 3y = 9`  and `6x + y = 4`  intercept each other at `(3/16 ; 23/8).`

You need to find the distance between `(0,2) ` and `(3/16 ; 23/8), ` hence you need to use the formula:

`D = sqrt((3/16 - 0)^2 + (23/8 - 2)^2)`

`D = sqrt(9/256 + 49/64) =gt D = sqrt(205/256) =gt D = sqrt(205)/16`

Hence, evaluating  the distance between the point of intesection between lines and the point (0,2) yields `D = sqrt(205)/16.`