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Solve the following systems of nonlinear equations.

3. 4x² + y² = 13
x² + y² = 10

[tex]\begin{cases}x = 1 \\ y = 3 \end{cases}[/tex] or [tex] \begin{cases}x = 1 \\ y = - 3 \end{cases}[/tex] or [tex] \begin{cases}x = - 1 \\ y = 3 \end{cases}[/tex] or [tex] \begin{cases} x= - 1 \\ y = - 3 \end{cases}[/tex]

Step-by-step explanation:

[tex]\begin{cases} {4x}^{2} + {y}^{2} = 13 \\ {x}^{2} + {y}^{2} = 10 \end{cases}[/tex]

Rearrange like terms to the same side of the equation

[tex] \begin{cases} {y}^{2} = 13 - {4x}^{2} \\ {x}^{2} + {y}^{2} = 10\end{cases}[/tex]

Substitute into one of the equations

[tex] {x}^{2} + 13 - {4x}^{2} = 10[/tex]

Rearrange all nonzero terms to the left side of the equation

[tex] {x}^{2} + 13 - {4x}^{2} - 10 = 0[/tex]

Combine like terms

[tex] { - 3x}^{2} + 3 = 0[/tex]

Reduce the greatest common factor on both sides of the equation

[tex] { - x}^{2} + 1 = 0[/tex]

Rearrange unknown terms to the left side of the equation

[tex] { - x}^{2} = - 1[/tex]

Divide both sides of the equation by the coefficient of the variable

[tex] {x}^{2} = 1[/tex]

Split into two equations

[tex]x = ± \sqrt{1} [/tex]

Combine the results

[tex]x = 1, \: x = - 1[/tex]

[tex] \begin{cases} {x}^{2} + {y}^{2} = 10 \\ x = 1 \end{cases}[/tex]

Substitute into one of the equations

[tex] {1}^{2} + {y}^{2} = 10[/tex]

Calculate the power

[tex]1 + {y}^{2} = 10[/tex]

Rearrange all nonzero terms to the left side of the equation

[tex]1 + {y}^{2} - 10 = 0[/tex]

Combine like terms

[tex] - 9 + {y}^{2} = 0[/tex]

Rearrange unknown terms to the left side of the equation

[tex] {y}^{2} = 9[/tex]

Split into two equations

[tex]y = ± \sqrt{9} [/tex]

Combine the results

[tex]y=3[/tex] or [tex] y=-3[/tex]

Same process for the [tex] \begin{cases} {x}^{2} + {y}^{2} = 10 \\ x = - 1 \end{cases} [/tex]

Therefore the solution of the system is

(y = 3 or y = -3)

[tex]\begin{cases}x = 1 \\ y = 3 \end{cases}[/tex] or [tex] \begin{cases}x = 1 \\ y = - 3 \end{cases}[/tex] or [tex] \begin{cases}x = - 1 \\ y = 3 \end{cases}[/tex] or [tex] \begin{cases} x= - 1 \\ y = - 3 \end{cases}[/tex]

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