Question

Prove that y=3x+2 has intersection with y=x-(1/x^2)+5 ?

Answer 1

They do intersect. Set them equal to each other with their common variable (x or y).If they have solutions then they intersect.

Explanation:

By setting the two equations equal to each other, you can verify if they intersect by simplifying and solving the equation. If the expression has a solution, there is an intersection and vise versa.

At their intersection, the point (x,y) is a valid input in both equations to make them true. Since x is the same in both equations at this point (or y), we can use this information to set them equal to each other (more accurately, substitute the y or x of one equation with the expression it is equal to in the other equation).

`y=3x+2`

`y=x-(1/x^2)+5`

`3x+2=x-(1/x^2)+5`

`2x-3=-(1/x^2)`

`[2x-3]*x^2=[-(1/x^2)]*x^2`

`2x^3-3x^2=-1`

`2x^3-3x^2+1=0`

`(2x+1)*(x-1)^2=0`

There are solutions for x meaning there are intersections in the two graphs.

Answer 2

A graphical plot shows that there is ONE intersection point near x = -3.5.

Explanation:

It might be proven by finding the solution as a set of linear equations. I could be simpler to just plot the two curves to indicate whether they have an intersection without obtaining the exact numerical solution.

`y = 3x + 2`
`y = x - (1/x^2) + 5`

`y - 3x = 2`
`y - x + (1/x^2) = 5` Subtract from the first equation:

`-2x - (1/x^2) = -3` ; `-2x^3 - 1 = -3x^2`
`-2x^3 + 3x^2 - 1 = 0`

GRAPHICALLY: