Solve {l}{x=sqrt{2}}{2x+8/x^2+4x+7=y}

$$\left\{ \begin{array} { l } { x = \sqrt { 2 } } \\ { \frac { 2 x + 8 } { x ^ { 2 } + 4 x + 7 } = y } \end{array} \right.$$

Solve for x, y

$$x=\sqrt{2}\approx 1.414213562$$

$$y=\frac{8-2\sqrt{2}}{7}\approx 0.738796125$$

Solution Steps

Consider the second equation. Insert the known values of variables into the equation.

$$\frac{2\sqrt{2}+8}{\left(\sqrt{2}\right)^{2}+4\sqrt{2}+7}=y$$

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{2}+8}{2+4\sqrt{2}+7}=y$$

Add 2 and 7 to get 9.

$$\frac{2\sqrt{2}+8}{9+4\sqrt{2}}=y$$

Rationalize the denominator of \frac{2\sqrt{2}+8}{9+4\sqrt{2}} by multiplying numerator and denominator by 9-4\sqrt{2}.

$$\frac{\left(2\sqrt{2}+8\right)\left(9-4\sqrt{2}\right)}{\left(9+4\sqrt{2}\right)\left(9-4\sqrt{2}\right)}=y$$

Consider \left(9+4\sqrt{2}\right)\left(9-4\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2\sqrt{2}+8\right)\left(9-4\sqrt{2}\right)}{9^{2}-\left(4\sqrt{2}\right)^{2}}=y$$

Calculate 9 to the power of 2 and get 81.

$$\frac{\left(2\sqrt{2}+8\right)\left(9-4\sqrt{2}\right)}{81-\left(4\sqrt{2}\right)^{2}}=y$$

Expand \left(4\sqrt{2}\right)^{2}.

$$\frac{\left(2\sqrt{2}+8\right)\left(9-4\sqrt{2}\right)}{81-4^{2}\left(\sqrt{2}\right)^{2}}=y$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(2\sqrt{2}+8\right)\left(9-4\sqrt{2}\right)}{81-16\left(\sqrt{2}\right)^{2}}=y$$

The square of \sqrt{2} is 2.

$$\frac{\left(2\sqrt{2}+8\right)\left(9-4\sqrt{2}\right)}{81-16\times 2}=y$$

Multiply 16 and 2 to get 32.

$$\frac{\left(2\sqrt{2}+8\right)\left(9-4\sqrt{2}\right)}{81-32}=y$$

Subtract 32 from 81 to get 49.

$$\frac{\left(2\sqrt{2}+8\right)\left(9-4\sqrt{2}\right)}{49}=y$$

Use the distributive property to multiply 2\sqrt{2}+8 by 9-4\sqrt{2} and combine like terms.

$$\frac{-14\sqrt{2}-8\left(\sqrt{2}\right)^{2}+72}{49}=y$$

The square of \sqrt{2} is 2.

$$\frac{-14\sqrt{2}-8\times 2+72}{49}=y$$

Multiply -8 and 2 to get -16.

$$\frac{-14\sqrt{2}-16+72}{49}=y$$

Add -16 and 72 to get 56.

$$\frac{-14\sqrt{2}+56}{49}=y$$

Divide each term of -14\sqrt{2}+56 by 49 to get -\frac{2}{7}\sqrt{2}+\frac{8}{7}.

$$-\frac{2}{7}\sqrt{2}+\frac{8}{7}=y$$

Swap sides so that all variable terms are on the left hand side.

$$y=-\frac{2}{7}\sqrt{2}+\frac{8}{7}$$

The system is now solved.

$$x=\sqrt{2} y=-\frac{2}{7}\sqrt{2}+\frac{8}{7}$$