x⋅y=2x-3y-1

Add 3y to both sides of the equation.

xy+3y=2x-1

Factor y out of xy+3y.

Factor y out of xy.

yx+3y=2x-1

Factor y out of 3y.

yx+y⋅3=2x-1

Factor y out of yx+y⋅3.

y(x+3)=2x-1

y(x+3)=2x-1

Divide each term by x+3 and simplify.

Divide each term in y(x+3)=2x-1 by x+3.

y(x+3)x+3=2xx+3+-1x+3

Cancel the common factor of x+3.

Cancel the common factor.

y(x+3)x+3=2xx+3+-1x+3

Divide y by 1.

y=2xx+3+-1x+3

y=2xx+3+-1x+3

Simplify 2xx+3+-1x+3.

Move the negative in front of the fraction.

y=2xx+3-1x+3

Combine the numerators over the common denominator.

y=2x-1x+3

y=2x-1x+3

y=2x-1x+3

y=2x-1x+3

Find where the expression 2x-1x+3 is undefined.

x=-3

Consider the rational function R(x)=axnbxm where n is the degree of the numerator and m is the degree of the denominator.

1. If n<m, then the x-axis, y=0, is the horizontal asymptote.

2. If n=m, then the horizontal asymptote is the line y=ab.

3. If n>m, then there is no horizontal asymptote (there is an oblique asymptote).

Find n and m.

n=1

m=1

Since n=m, the horizontal asymptote is the line y=ab where a=2 and b=1.

y=2

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

This is the set of all asymptotes.

Vertical Asymptotes: x=-3

Horizontal Asymptotes: y=2

No Oblique Asymptotes

Vertical Asymptotes: x=-3

Horizontal Asymptotes: y=2

No Oblique Asymptotes

Graph x*y=2x-3y-1