Question

Need help with a system of equations

Answer 1

Answer:

$\boxed{x - y = -2\sqrt{7}}$

Given:

$x + y = 4 \\ \\ {x}^{2} + {y}^{2} = 22 \\ \\ {x}^{4} = {y}^{4} - 176 \sqrt{7}$

Step-by-step explanation:

$= > {x}^{4} = {y}^{4} - 176 \sqrt{7} \\ \\ = > {x}^{4} - {y}^{4} = - 176 \sqrt{7} \\ \\ = > {( {x}^{2}) }^{2} - { ({y}^{2}) }^{2} = - 176 \sqrt{7} \\ \\ = > ( {x}^{2} + {y}^{2} )( {x}^{2} - {y}^{2} ) = - 176 \sqrt{7} \\ \\ = > ( {x}^{2} + {y}^{2}) (x + y)(x - y) = - 176 \sqrt{?} \\ \\ = > 22 \times 4(x - y) = - 176 \sqrt{7} \\ \\ = > 88(x - y) = - 176 \sqrt{7} \\ \\ = > x - y = - \frac{176 \sqrt{7} }{88} \\ \\ = > x - y = - 2 \sqrt{7}$

Identity used:

${x}^{2} - {y}^{2} = (x + y)(x - y)$

Answer 2

Answer:

-2$\sqrt{7}$

Step-by-step explanation:

$x^{4} - y^{4} = -176\sqrt{7} \\(x^{2} + y^{2})(x^{2} - y^{2} ) = -176\sqrt{7} \\(x^{2} - y^{2} )= \frac{-176\sqrt{7}}{22} \\(x + y)(x - y) = -8\sqrt{7}\\ x - y = \frac{-8\sqrt{7}}{4} = -2\sqrt{7}$