What is the coordinates of the point of intersection of the curve y=1x+2+3 and the line y=x?

1 Answer
Jan 18, 2018

Those two curves do not intersect.

Explanation:

To find an intersection we try to find an x-value where both equations give us the same y value.

So we want to solve:

1x+2+3=x

We'll begin by clearing the fraction. Multiply both sides of the equation by (x+2)

(x+2)(1x+2+3)=x(x+2)

1+3(x+2)=x(x+2)

1+3x+6=x22x

We see an x2, so this is a qudratic equation. The usual way of writing a quadratic equation is to have all of the terms on one side equal to 0 on the other, so let's make it like that.

x2+5x+7=0.

We try to solve by factoring, but we don't see how to factor this, so either solve by completing the square or solve using the quadratic formula.

x=b±b24ac2a

x=(5)±(5)24(1)(7)2(1)

But now we notice (if we didn't notice before) that

b24ac=(5)24(1)(7)=2528=3.

So there is no solution in the real numbers.

Therefore, there is no point of intersection for the curves.

Here is the graph. Of course, y=x is the straight line.

graph{(y+x)(y-1/(x+2)-3)=0 [-17.02, 15.01, -5.2, 10.82]}