Sure, let's go through the steps to solve the equation [tex]\(3x^2 + 3y^2 = 0\)[/tex]:
1. Divide the Equation by 3:
[tex]\[
3x^2 + 3y^2 = 0
\][/tex]
Dividing both sides of the equation by 3 simplifies the equation:
[tex]\[
x^2 + y^2 = 0
\][/tex]
2. Analyze the Equation [tex]\(x^2 + y^2 = 0\)[/tex]:
Notice that [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex] are both squared terms, which means they are both non-negative (i.e., [tex]\(x^2 \geq 0\)[/tex] and [tex]\(y^2 \geq 0\)[/tex]). The sum of two non-negative terms equals zero only if both terms are zero.
3. Set Each Term to Zero:
Therefore, for the equation [tex]\(x^2 + y^2 = 0\)[/tex] to hold true, both:
[tex]\[
x^2 = 0
\][/tex]
and
[tex]\[
y^2 = 0
\][/tex]
4. Solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
If [tex]\(x^2 = 0\)[/tex], then the only solution is:
[tex]\[
x = 0
\][/tex]
Similarly, if [tex]\(y^2 = 0\)[/tex], then the only solution is:
[tex]\[
y = 0
\][/tex]
Therefore, the solution to the equation [tex]\(3x^2 + 3y^2 = 0\)[/tex] is:
[tex]\[
(x, y) = (0, 0)
\][/tex]
This means both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] must be zero to satisfy the original equation.
