Answer :
To solve the given system of nonlinear equations:
[tex]\[
\begin{align*}
1. & \quad 7x^2 + 7y^2 = 84 \\
2. & \quad 35x^2 + 35y^2 = 336 \\
\end{align*}
\][/tex]
First, notice that both equations can be simplified by dividing each term by their coefficients.
For the first equation:
[tex]\[
7x^2 + 7y^2 = 84
\][/tex]
Divide every term by 7:
[tex]\[
x^2 + y^2 = 12
\][/tex]
For the second equation:
[tex]\[
35x^2 + 35y^2 = 336
\][/tex]
Divide every term by 35:
[tex]\[
x^2 + y^2 = \frac{336}{35}
\][/tex]
Calculate the right side:
[tex]\[
x^2 + y^2 = \frac{336}{35} = 9.6
\][/tex]
Now we have a simpler system:
[tex]\[
\begin{align*}
1. & \quad x^2 + y^2 = 12 \\
2. & \quad x^2 + y^2 = 9.6 \\
\end{align*}
\][/tex]
Notice that the left-hand sides of both equations are the same, while the right-hand sides differ.
This situation indicates a contradiction: the same expression [tex]\(x^2 + y^2\)[/tex] cannot equal both 12 and 9.6 simultaneously. Therefore, there is no solution that satisfies both equations at the same time.
Thus, the system of equations does not have any real solutions.
[tex]\[
\begin{align*}
1. & \quad 7x^2 + 7y^2 = 84 \\
2. & \quad 35x^2 + 35y^2 = 336 \\
\end{align*}
\][/tex]
First, notice that both equations can be simplified by dividing each term by their coefficients.
For the first equation:
[tex]\[
7x^2 + 7y^2 = 84
\][/tex]
Divide every term by 7:
[tex]\[
x^2 + y^2 = 12
\][/tex]
For the second equation:
[tex]\[
35x^2 + 35y^2 = 336
\][/tex]
Divide every term by 35:
[tex]\[
x^2 + y^2 = \frac{336}{35}
\][/tex]
Calculate the right side:
[tex]\[
x^2 + y^2 = \frac{336}{35} = 9.6
\][/tex]
Now we have a simpler system:
[tex]\[
\begin{align*}
1. & \quad x^2 + y^2 = 12 \\
2. & \quad x^2 + y^2 = 9.6 \\
\end{align*}
\][/tex]
Notice that the left-hand sides of both equations are the same, while the right-hand sides differ.
This situation indicates a contradiction: the same expression [tex]\(x^2 + y^2\)[/tex] cannot equal both 12 and 9.6 simultaneously. Therefore, there is no solution that satisfies both equations at the same time.
Thus, the system of equations does not have any real solutions.
