Solve {l}{x+y+z=15quad1/61quadsqrt{y}quadxy+yz+xz=45}{sqrt{a}quadx^2+y^2+z^2=?}

$$\left. \begin{array} { l } { x + y + z = 15 \quad \frac { 1 } { 61 } \quad \sqrt { y } \quad x y + y z + x z = 45 } \\ { \sqrt { a } \quad x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = ? } \end{array} \right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{61\left(z+y-yz\right)}{61-61z-916y^{\frac{3}{2}}}\text{, }&\left(\frac{61-61z}{916}<0\text{ or }y\neq \frac{\left(6397802-6397802z\right)^{\frac{2}{3}}}{209764}\right)\text{ and }y\geq 0\\x\in \mathrm{R}\text{, }&\frac{61-61z}{916}\geq 0\text{ and }\left(z\leq 0\text{ or }z>1\right)\text{ and }y=-\frac{z}{1-z}\text{ and }-\frac{z}{1-z}=\frac{\left(6397802-6397802z\right)^{\frac{2}{3}}}{209764}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 61.

$$61x+61y+61z=\left(15\times 61+1\right)\sqrt{y}xy+61yz+61xz$$

Multiply 15 and 61 to get 915.

$$61x+61y+61z=\left(915+1\right)\sqrt{y}xy+61yz+61xz$$

Add 915 and 1 to get 916.

$$61x+61y+61z=916\sqrt{y}xy+61yz+61xz$$

Subtract 916\sqrt{y}xy from both sides.

$$61x+61y+61z-916\sqrt{y}xy=61yz+61xz$$

Subtract 61xz from both sides.

$$61x+61y+61z-916\sqrt{y}xy-61xz=61yz$$

Subtract 61y from both sides.

$$61x+61z-916\sqrt{y}xy-61xz=61yz-61y$$

Subtract 61z from both sides.

$$61x-916\sqrt{y}xy-61xz=61yz-61y-61z$$

Combine all terms containing x.

$$\left(61-916\sqrt{y}y-61z\right)x=61yz-61y-61z$$

The equation is in standard form.

$$\left(-916\sqrt{y}y-61z+61\right)x=61yz-61y-61z$$

Divide both sides by 61-916\sqrt{y}y-61z.

$$\frac{\left(-916\sqrt{y}y-61z+61\right)x}{-916\sqrt{y}y-61z+61}=\frac{61yz-61y-61z}{-916\sqrt{y}y-61z+61}$$

Dividing by 61-916\sqrt{y}y-61z undoes the multiplication by 61-916\sqrt{y}y-61z.

$$x=\frac{61yz-61y-61z}{-916\sqrt{y}y-61z+61}$$

Divide 61yz-61y-61z by 61-916\sqrt{y}y-61z.

$$x=\frac{61\left(yz-y-z\right)}{61-61z-916y^{\frac{3}{2}}}$$