Sure, let's solve the equation step-by-step.
1. Given Equation:
[tex]\[
6xy^2 - 12x^2y = 6xy
\][/tex]
2. Move all terms to one side to set the equation to 0:
[tex]\[
6xy^2 - 12x^2y - 6xy = 0
\][/tex]
3. Factor out the common term on the left side:
Notice that [tex]\(6xy\)[/tex] is a common factor for all the terms.
[tex]\[
6xy(y - 2x - 1) = 0
\][/tex]
4. Set each factor equal to zero:
The product of terms is zero if and only if at least one of the terms is zero.
[tex]\[
6xy = 0 \quad \text{or} \quad y - 2x - 1 = 0
\][/tex]
5. Solve the first factor (6xy = 0):
For [tex]\(6xy = 0\)[/tex], either [tex]\(x = 0\)[/tex] or [tex]\(y = 0\)[/tex].
- If [tex]\(x = 0\)[/tex], the equation is satisfied regardless of [tex]\(y\)[/tex].
- If [tex]\(y = 0\)[/tex], the equation is satisfied regardless of [tex]\(x\)[/tex].
6. Solve the second factor (y - 2x - 1 = 0):
[tex]\[
y - 2x - 1 = 0 \implies y = 2x + 1
\][/tex]
Summary of Solutions:
- [tex]\(x = 0\)[/tex]
- [tex]\(y = 0\)[/tex]
- [tex]\(y = 2x + 1\)[/tex]
These solutions describe all possible pairs [tex]\((x, y)\)[/tex] that satisfy the original equation.
