Linear Algebra Examples

$[\begin{array}{c} - 4 & \sqrt{6} & \sqrt{2} \\ \sqrt{6} & - 3 & \sqrt{3} \\ \sqrt{2} & \sqrt{3} & - 5 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1

Multiply $[\begin{array}{c} - 4 & \sqrt{6} & \sqrt{2} \\ \sqrt{6} & - 3 & \sqrt{3} \\ \sqrt{2} & \sqrt{3} & - 5 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}]$ .

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Step 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $3 \times 3$ and the second matrix is $3 \times 1$ .

Step 1.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} - 4 x + \sqrt{6} y + \sqrt{2} z \\ \sqrt{6} x - 3 y + \sqrt{3} z \\ \sqrt{2} x + \sqrt{3} y - 5 z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 2

Write as a linear system of equations.

$- 4 x + \sqrt{6} y + \sqrt{2} z = 0$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $x$ in $- 4 x + \sqrt{6} y + \sqrt{2} z = 0$ .

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Step 3.1.1

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1.1.1

Subtract $\sqrt{6} y$ from both sides of the equation.

$- 4 x + \sqrt{2} z = - \sqrt{6} y$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.1.1.2

Subtract $\sqrt{2} z$ from both sides of the equation.

$- 4 x = - \sqrt{6} y - \sqrt{2} z$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$- 4 x = - \sqrt{6} y - \sqrt{2} z$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.1.2

Divide each term in $- 4 x = - \sqrt{6} y - \sqrt{2} z$ by $- 4$ and simplify.

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Step 3.1.2.1

Divide each term in $- 4 x = - \sqrt{6} y - \sqrt{2} z$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- \sqrt{6} y}{- 4} + \frac{- \sqrt{2} z}{- 4}$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- \sqrt{6} y}{- 4} + \frac{- \sqrt{2} z}{- 4}$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \sqrt{6} y}{- 4} + \frac{- \sqrt{2} z}{- 4}$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$x = \frac{- \sqrt{6} y}{- 4} + \frac{- \sqrt{2} z}{- 4}$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$x = \frac{- \sqrt{6} y}{- 4} + \frac{- \sqrt{2} z}{- 4}$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Simplify each term.

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Step 3.1.2.3.1.1

Dividing two negative values results in a positive value.

$x = \frac{\sqrt{6} y}{4} + \frac{- \sqrt{2} z}{- 4}$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.1.2.3.1.2

Dividing two negative values results in a positive value.

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{6} x - 3 y + \sqrt{3} z = 0$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2

Replace all occurrences of $x$ with $\frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$ in each equation.

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Step 3.2.1

Replace all occurrences of $x$ in $\sqrt{6} x - 3 y + \sqrt{3} z = 0$ with $\frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$ .

$\sqrt{6} (\frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}) - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify $\sqrt{6} (\frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}) - 3 y + \sqrt{3} z$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$\sqrt{6} (\frac{\sqrt{6} y}{4}) + \sqrt{6} (\frac{\sqrt{2} z}{4}) - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.2

Multiply $\sqrt{6} \frac{\sqrt{6} y}{4}$ .

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Step 3.2.2.1.1.2.1

Combine $\sqrt{6}$ and $\frac{\sqrt{6} y}{4}$ .

$\frac{\sqrt{6} (\sqrt{6} y)}{4} + \sqrt{6} (\frac{\sqrt{2} z}{4}) - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.2.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{6} \sqrt{6} y}{4} + \sqrt{6} (\frac{\sqrt{2} z}{4}) - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.2.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{6} \sqrt{6} y}{4} + \sqrt{6} (\frac{\sqrt{2} z}{4}) - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{\sqrt{6}}^{1 + 1} y}{4} + \sqrt{6} (\frac{\sqrt{2} z}{4}) - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.2.5

Add $1$ and $1$ .

$\frac{{\sqrt{6}}^{2} y}{4} + \sqrt{6} (\frac{\sqrt{2} z}{4}) - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{{\sqrt{6}}^{2} y}{4} + \sqrt{6} (\frac{\sqrt{2} z}{4}) - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.3

Multiply $\sqrt{6} \frac{\sqrt{2} z}{4}$ .

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Step 3.2.2.1.1.3.1

Combine $\sqrt{6}$ and $\frac{\sqrt{2} z}{4}$ .

$\frac{{\sqrt{6}}^{2} y}{4} + \frac{\sqrt{6} (\sqrt{2} z)}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.3.2

Combine using the product rule for radicals.

$\frac{{\sqrt{6}}^{2} y}{4} + \frac{\sqrt{2 \cdot 6} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.3.3

Multiply $2$ by $6$ .

$\frac{{\sqrt{6}}^{2} y}{4} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{{\sqrt{6}}^{2} y}{4} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4

Simplify each term.

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Step 3.2.2.1.1.4.1

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.2.2.1.1.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{{(6^{\frac{1}{2}})}^{2} y}{4} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{6^{\frac{1}{2} \cdot 2} y}{4} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{6^{\frac{2}{2}} y}{4} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.1.4

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.4.1.4.1

Cancel the common factor.

$\frac{6^{\frac{2}{2}} y}{4} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.1.4.2

Rewrite the expression.

$\frac{6 y}{4} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{6 y}{4} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.1.5

Evaluate the exponent.

$\frac{6 y}{4} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{6 y}{4} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.2

Cancel the common factor of $6$ and $4$ .

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Step 3.2.2.1.1.4.2.1

Factor $2$ out of $6 y$ .

$\frac{2 (3 y)}{4} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.2.2

Cancel the common factors.

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Step 3.2.2.1.1.4.2.2.1

Factor $2$ out of $4$ .

$\frac{2 (3 y)}{2 (2)} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.2.2.2

Cancel the common factor.

$\frac{2 (3 y)}{2 \cdot 2} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.2.2.3

Rewrite the expression.

$\frac{3 y}{2} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{3 y}{2} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{3 y}{2} + \frac{\sqrt{12} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.3

Simplify the numerator.

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Step 3.2.2.1.1.4.3.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 3.2.2.1.1.4.3.1.1

Factor $4$ out of $12$ .

$\frac{3 y}{2} + \frac{\sqrt{4 (3)} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.3.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{3 y}{2} + \frac{\sqrt{2^{2} \cdot 3} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{3 y}{2} + \frac{\sqrt{2^{2} \cdot 3} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.3.2

Pull terms out from under the radical.

$\frac{3 y}{2} + \frac{2 \sqrt{3} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{3 y}{2} + \frac{2 \sqrt{3} z}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.4

Cancel the common factor of $2$ and $4$ .

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Step 3.2.2.1.1.4.4.1

Factor $2$ out of $2 \sqrt{3} z$ .

$\frac{3 y}{2} + \frac{2 (\sqrt{3} z)}{4} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.4.2

Cancel the common factors.

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Step 3.2.2.1.1.4.4.2.1

Factor $2$ out of $4$ .

$\frac{3 y}{2} + \frac{2 (\sqrt{3} z)}{2 \cdot 2} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.4.2.2

Cancel the common factor.

$\frac{3 y}{2} + \frac{2 (\sqrt{3} z)}{2 \cdot 2} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.1.4.4.2.3

Rewrite the expression.

$\frac{3 y}{2} + \frac{\sqrt{3} z}{2} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{3 y}{2} + \frac{\sqrt{3} z}{2} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{3 y}{2} + \frac{\sqrt{3} z}{2} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{3 y}{2} + \frac{\sqrt{3} z}{2} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{3 y}{2} + \frac{\sqrt{3} z}{2} - 3 y + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.2

To write $- 3 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sqrt{3} z}{2} + \frac{3 y}{2} - 3 y \cdot \frac{2}{2} + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.3

Combine $- 3 y$ and $\frac{2}{2}$ .

$\frac{\sqrt{3} z}{2} + \frac{3 y}{2} + \frac{- 3 y \cdot 2}{2} + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.4

Combine the numerators over the common denominator.

$\frac{\sqrt{3} z}{2} + \frac{3 y - 3 y \cdot 2}{2} + \sqrt{3} z = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.5

Combine the numerators over the common denominator.

$\sqrt{3} z + \frac{\sqrt{3} z + 3 y - 3 y \cdot 2}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.6

Multiply $2$ by $- 3$ .

$\sqrt{3} z + \frac{\sqrt{3} z + 3 y - 6 y}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.7

Subtract $6 y$ from $3 y$ .

$\sqrt{3} z + \frac{\sqrt{3} z - 3 y}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.8

To write $\sqrt{3} z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{3} z \cdot \frac{2}{2} + \frac{\sqrt{3} z - 3 y}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.9

Simplify terms.

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Step 3.2.2.1.9.1

Combine $\sqrt{3} z$ and $\frac{2}{2}$ .

$\frac{\sqrt{3} z \cdot 2}{2} + \frac{\sqrt{3} z - 3 y}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.9.2

Combine the numerators over the common denominator.

$\frac{\sqrt{3} z \cdot 2 + \sqrt{3} z - 3 y}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{\sqrt{3} z \cdot 2 + \sqrt{3} z - 3 y}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.10

Simplify the numerator.

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Step 3.2.2.1.10.1

Move $2$ to the left of $\sqrt{3} z$ .

$\frac{2 \cdot (\sqrt{3} z) + \sqrt{3} z - 3 y}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.10.2

Add $2 \sqrt{3} z$ and $\sqrt{3} z$ .

$\frac{3 \sqrt{3} z - 3 y}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.10.3

Factor $3$ out of $3 \sqrt{3} z - 3 y$ .

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Step 3.2.2.1.10.3.1

Factor $3$ out of $3 \sqrt{3} z$ .

$\frac{3 (\sqrt{3} z) - 3 y}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.10.3.2

Factor $3$ out of $- 3 y$ .

$\frac{3 (\sqrt{3} z) + 3 (- y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.2.1.10.3.3

Factor $3$ out of $3 (\sqrt{3} z) + 3 (- y)$ .

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\sqrt{2} x + \sqrt{3} y - 5 z = 0$

Step 3.2.3

Replace all occurrences of $x$ in $\sqrt{2} x + \sqrt{3} y - 5 z = 0$ with $\frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$ .

$\sqrt{2} (\frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}) + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4

Simplify the left side.

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Step 3.2.4.1

Simplify $\sqrt{2} (\frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}) + \sqrt{3} y - 5 z$ .

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Step 3.2.4.1.1

Simplify each term.

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Step 3.2.4.1.1.1

Apply the distributive property.

$\sqrt{2} (\frac{\sqrt{6} y}{4}) + \sqrt{2} (\frac{\sqrt{2} z}{4}) + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.2

Multiply $\sqrt{2} \frac{\sqrt{6} y}{4}$ .

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Step 3.2.4.1.1.2.1

Combine $\sqrt{2}$ and $\frac{\sqrt{6} y}{4}$ .

$\frac{\sqrt{2} (\sqrt{6} y)}{4} + \sqrt{2} (\frac{\sqrt{2} z}{4}) + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.2.2

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 2} y}{4} + \sqrt{2} (\frac{\sqrt{2} z}{4}) + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.2.3

Multiply $6$ by $2$ .

$\frac{\sqrt{12} y}{4} + \sqrt{2} (\frac{\sqrt{2} z}{4}) + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{\sqrt{12} y}{4} + \sqrt{2} (\frac{\sqrt{2} z}{4}) + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.3

Multiply $\sqrt{2} \frac{\sqrt{2} z}{4}$ .

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Step 3.2.4.1.1.3.1

Combine $\sqrt{2}$ and $\frac{\sqrt{2} z}{4}$ .

$\frac{\sqrt{12} y}{4} + \frac{\sqrt{2} (\sqrt{2} z)}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{12} y}{4} + \frac{\sqrt{2} \sqrt{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{12} y}{4} + \frac{\sqrt{2} \sqrt{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{12} y}{4} + \frac{{\sqrt{2}}^{1 + 1} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.3.5

Add $1$ and $1$ .

$\frac{\sqrt{12} y}{4} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{\sqrt{12} y}{4} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4

Simplify each term.

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Step 3.2.4.1.1.4.1

Simplify the numerator.

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Step 3.2.4.1.1.4.1.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 3.2.4.1.1.4.1.1.1

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)} y}{4} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.1.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 3} y}{4} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{\sqrt{2^{2} \cdot 3} y}{4} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.1.2

Pull terms out from under the radical.

$\frac{2 \sqrt{3} y}{4} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{2 \sqrt{3} y}{4} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.2

Cancel the common factor of $2$ and $4$ .

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Step 3.2.4.1.1.4.2.1

Factor $2$ out of $2 \sqrt{3} y$ .

$\frac{2 (\sqrt{3} y)}{4} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.2.2

Cancel the common factors.

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Step 3.2.4.1.1.4.2.2.1

Factor $2$ out of $4$ .

$\frac{2 (\sqrt{3} y)}{2 \cdot 2} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.2.2.2

Cancel the common factor.

$\frac{2 (\sqrt{3} y)}{2 \cdot 2} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.2.2.3

Rewrite the expression.

$\frac{\sqrt{3} y}{2} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{\sqrt{3} y}{2} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{\sqrt{3} y}{2} + \frac{{\sqrt{2}}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.3

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.2.4.1.1.4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\sqrt{3} y}{2} + \frac{{(2^{\frac{1}{2}})}^{2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{3} y}{2} + \frac{2^{\frac{1}{2} \cdot 2} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.3.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{3} y}{2} + \frac{2^{\frac{2}{2}} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.3.4

Cancel the common factor of $2$ .

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Step 3.2.4.1.1.4.3.4.1

Cancel the common factor.

$\frac{\sqrt{3} y}{2} + \frac{2^{\frac{2}{2}} z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.3.4.2

Rewrite the expression.

$\frac{\sqrt{3} y}{2} + \frac{2 z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{\sqrt{3} y}{2} + \frac{2 z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.3.5

Evaluate the exponent.

$\frac{\sqrt{3} y}{2} + \frac{2 z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{\sqrt{3} y}{2} + \frac{2 z}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.4

Cancel the common factor of $2$ and $4$ .

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Step 3.2.4.1.1.4.4.1

Factor $2$ out of $2 z$ .

$\frac{\sqrt{3} y}{2} + \frac{2 (z)}{4} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.4.2

Cancel the common factors.

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Step 3.2.4.1.1.4.4.2.1

Factor $2$ out of $4$ .

$\frac{\sqrt{3} y}{2} + \frac{2 z}{2 \cdot 2} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.4.2.2

Cancel the common factor.

$\frac{\sqrt{3} y}{2} + \frac{2 z}{2 \cdot 2} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.1.4.4.2.3

Rewrite the expression.

$\frac{\sqrt{3} y}{2} + \frac{z}{2} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{\sqrt{3} y}{2} + \frac{z}{2} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{\sqrt{3} y}{2} + \frac{z}{2} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{\sqrt{3} y}{2} + \frac{z}{2} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{\sqrt{3} y}{2} + \frac{z}{2} + \sqrt{3} y - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.2

To write $\sqrt{3} y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{z}{2} + \frac{\sqrt{3} y}{2} + \sqrt{3} y \cdot \frac{2}{2} - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.3

Simplify terms.

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Step 3.2.4.1.3.1

Combine $\sqrt{3} y$ and $\frac{2}{2}$ .

$\frac{z}{2} + \frac{\sqrt{3} y}{2} + \frac{\sqrt{3} y \cdot 2}{2} - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.3.2

Combine the numerators over the common denominator.

$\frac{z}{2} + \frac{\sqrt{3} y + \sqrt{3} y \cdot 2}{2} - 5 z = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.3.3

Combine the numerators over the common denominator.

$- 5 z + \frac{z + \sqrt{3} y + \sqrt{3} y \cdot 2}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$- 5 z + \frac{z + \sqrt{3} y + \sqrt{3} y \cdot 2}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.4

Move $2$ to the left of $\sqrt{3} y$ .

$- 5 z + \frac{z + \sqrt{3} y + 2 \sqrt{3} y}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.5

Add $\sqrt{3} y$ and $2 \sqrt{3} y$ .

$- 5 z + \frac{z + 3 \sqrt{3} y}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.6

To write $- 5 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 5 z \cdot \frac{2}{2} + \frac{z + 3 \sqrt{3} y}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.7

Simplify terms.

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Step 3.2.4.1.7.1

Combine $- 5 z$ and $\frac{2}{2}$ .

$\frac{- 5 z \cdot 2}{2} + \frac{z + 3 \sqrt{3} y}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.7.2

Combine the numerators over the common denominator.

$\frac{- 5 z \cdot 2 + z + 3 \sqrt{3} y}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{- 5 z \cdot 2 + z + 3 \sqrt{3} y}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.8

Simplify the numerator.

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Step 3.2.4.1.8.1

Multiply $2$ by $- 5$ .

$\frac{- 10 z + z + 3 \sqrt{3} y}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.8.2

Add $- 10 z$ and $z$ .

$\frac{- 9 z + 3 \sqrt{3} y}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.8.3

Factor $3$ out of $- 9 z + 3 \sqrt{3} y$ .

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Step 3.2.4.1.8.3.1

Factor $3$ out of $- 9 z$ .

$\frac{3 (- 3 z) + 3 \sqrt{3} y}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.8.3.2

Factor $3$ out of $3 \sqrt{3} y$ .

$\frac{3 (- 3 z) + 3 (\sqrt{3} y)}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.8.3.3

Factor $3$ out of $3 (- 3 z) + 3 (\sqrt{3} y)$ .

$\frac{3 (- 3 z + \sqrt{3} y)}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{3 (- 3 z + \sqrt{3} y)}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{3 (- 3 z + \sqrt{3} y)}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.9

Simplify with factoring out.

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Step 3.2.4.1.9.1

Factor $- 1$ out of $- 3 z$ .

$\frac{3 (- (3 z) + \sqrt{3} y)}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.9.2

Factor $- 1$ out of $\sqrt{3} y$ .

$\frac{3 (- (3 z) - (- \sqrt{3} y))}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.9.3

Factor $- 1$ out of $- (3 z) - (- \sqrt{3} y)$ .

$\frac{3 (- (3 z - \sqrt{3} y))}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.9.4

Simplify the expression.

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Step 3.2.4.1.9.4.1

Rewrite $- (3 z - \sqrt{3} y)$ as $- 1 (3 z - \sqrt{3} y)$ .

$\frac{3 (- 1 (3 z - \sqrt{3} y))}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.2.4.1.9.4.2

Move the negative in front of the fraction.

$- \frac{3 (3 z - \sqrt{3} y)}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$- \frac{3 (3 z - \sqrt{3} y)}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$- \frac{3 (3 z - \sqrt{3} y)}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$- \frac{3 (3 z - \sqrt{3} y)}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$- \frac{3 (3 z - \sqrt{3} y)}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$- \frac{3 (3 z - \sqrt{3} y)}{2} = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.3

Solve for $z$ in $- \frac{3 (3 z - \sqrt{3} y)}{2} = 0$ .

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Step 3.3.1

Set the numerator equal to zero.

$3 (3 z - \sqrt{3} y) = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.3.2

Solve the equation for $z$ .

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Step 3.3.2.1

Divide each term in $3 (3 z - \sqrt{3} y) = 0$ by $3$ and simplify.

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Step 3.3.2.1.1

Divide each term in $3 (3 z - \sqrt{3} y) = 0$ by $3$ .

$\frac{3 (3 z - \sqrt{3} y)}{3} = \frac{0}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.3.2.1.2

Simplify the left side.

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Step 3.3.2.1.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.2.1.1

Cancel the common factor.

$\frac{3 (3 z - \sqrt{3} y)}{3} = \frac{0}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.3.2.1.2.1.2

Divide $3 z - \sqrt{3} y$ by $1$ .

$3 z - \sqrt{3} y = \frac{0}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$3 z - \sqrt{3} y = \frac{0}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$3 z - \sqrt{3} y = \frac{0}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.3.2.1.3

Simplify the right side.

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Step 3.3.2.1.3.1

Divide $0$ by $3$ .

$3 z - \sqrt{3} y = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$3 z - \sqrt{3} y = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$3 z - \sqrt{3} y = 0$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.3.2.2

Add $\sqrt{3} y$ to both sides of the equation.

$3 z = \sqrt{3} y$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.3.2.3

Divide each term in $3 z = \sqrt{3} y$ by $3$ and simplify.

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Step 3.3.2.3.1

Divide each term in $3 z = \sqrt{3} y$ by $3$ .

$\frac{3 z}{3} = \frac{\sqrt{3} y}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.3.2.3.2

Simplify the left side.

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Step 3.3.2.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.3.2.1.1

Cancel the common factor.

$\frac{3 z}{3} = \frac{\sqrt{3} y}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.3.2.3.2.1.2

Divide $z$ by $1$ .

$z = \frac{\sqrt{3} y}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$z = \frac{\sqrt{3} y}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$z = \frac{\sqrt{3} y}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$z = \frac{\sqrt{3} y}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$z = \frac{\sqrt{3} y}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$z = \frac{\sqrt{3} y}{3}$

$\frac{3 (\sqrt{3} z - y)}{2} = 0$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4

Replace all occurrences of $z$ with $\frac{\sqrt{3} y}{3}$ in each equation.

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Step 3.4.1

Replace all occurrences of $z$ in $\frac{3 (\sqrt{3} z - y)}{2} = 0$ with $\frac{\sqrt{3} y}{3}$ .

$\frac{3 (\sqrt{3} (\frac{\sqrt{3} y}{3}) - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify $\frac{3 (\sqrt{3} (\frac{\sqrt{3} y}{3}) - y)}{2}$ .

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Step 3.4.2.1.1

Simplify the numerator.

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Step 3.4.2.1.1.1

Multiply $\sqrt{3} \frac{\sqrt{3} y}{3}$ .

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Step 3.4.2.1.1.1.1

Combine $\sqrt{3}$ and $\frac{\sqrt{3} y}{3}$ .

$\frac{3 (\frac{\sqrt{3} (\sqrt{3} y)}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.1.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{3 (\frac{\sqrt{3} \sqrt{3} y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.1.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{3 (\frac{\sqrt{3} \sqrt{3} y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (\frac{{\sqrt{3}}^{1 + 1} y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.1.5

Add $1$ and $1$ .

$\frac{3 (\frac{{\sqrt{3}}^{2} y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{3 (\frac{{\sqrt{3}}^{2} y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.4.2.1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{3 (\frac{{(3^{\frac{1}{2}})}^{2} y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 (\frac{3^{\frac{1}{2} \cdot 2} y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 (\frac{3^{\frac{2}{2}} y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.2.4

Cancel the common factor of $2$ .

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Step 3.4.2.1.1.2.4.1

Cancel the common factor.

$\frac{3 (\frac{3^{\frac{2}{2}} y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.2.4.2

Rewrite the expression.

$\frac{3 (\frac{3 y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{3 (\frac{3 y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.2.5

Evaluate the exponent.

$\frac{3 (\frac{3 y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{3 (\frac{3 y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.3

Cancel the common factor of $3$ .

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Step 3.4.2.1.1.3.1

Cancel the common factor.

$\frac{3 (\frac{3 y}{3} - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.3.2

Divide $y$ by $1$ .

$\frac{3 (y - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{3 (y - y)}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.1.4

Subtract $y$ from $y$ .

$\frac{3 \cdot 0}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$\frac{3 \cdot 0}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.2

Simplify the expression.

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Step 3.4.2.1.2.1

Multiply $3$ by $0$ .

$\frac{0}{2} = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.2.1.2.2

Divide $0$ by $2$ .

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$

Step 3.4.3

Replace all occurrences of $z$ in $x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} z}{4}$ with $\frac{\sqrt{3} y}{3}$ .

$x = \frac{\sqrt{6} y}{4} + \frac{\sqrt{2} (\frac{\sqrt{3} y}{3})}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $\frac{\sqrt{6} y}{4} + \frac{\sqrt{2} (\frac{\sqrt{3} y}{3})}{4}$ .

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Step 3.4.4.1.1

Combine the numerators over the common denominator.

$x = \frac{\sqrt{6} y + \sqrt{2} (\frac{\sqrt{3} y}{3})}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.2

Multiply $\sqrt{2} (\frac{\sqrt{3} y}{3})$ .

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Step 3.4.4.1.2.1

Combine $\sqrt{2}$ and $\frac{\sqrt{3} y}{3}$ .

$x = \frac{\sqrt{6} y + \frac{\sqrt{2} (\sqrt{3} y)}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.2.2

Combine using the product rule for radicals.

$x = \frac{\sqrt{6} y + \frac{\sqrt{3 \cdot 2} y}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.2.3

Multiply $3$ by $2$ .

$x = \frac{\sqrt{6} y + \frac{\sqrt{6} y}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y + \frac{\sqrt{6} y}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.3

To write $\sqrt{6} y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{\sqrt{6} y \cdot \frac{3}{3} + \frac{\sqrt{6} y}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.4

Simplify terms.

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Step 3.4.4.1.4.1

Combine $\sqrt{6} y$ and $\frac{3}{3}$ .

$x = \frac{\frac{\sqrt{6} y \cdot 3}{3} + \frac{\sqrt{6} y}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.4.2

Combine the numerators over the common denominator.

$x = \frac{\frac{\sqrt{6} y \cdot 3 + \sqrt{6} y}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\frac{\sqrt{6} y \cdot 3 + \sqrt{6} y}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.5

Simplify the numerator.

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Step 3.4.4.1.5.1

Factor $\sqrt{6} y$ out of $\sqrt{6} y \cdot 3 + \sqrt{6} y$ .

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Step 3.4.4.1.5.1.1

Factor $\sqrt{6} y$ out of $\sqrt{6} y \cdot 3$ .

$x = \frac{\frac{\sqrt{6} y (3) + \sqrt{6} y}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.5.1.2

Factor $\sqrt{6} y$ out of $\sqrt{6} y$ .

$x = \frac{\frac{\sqrt{6} y (3) + \sqrt{6} y (1)}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.5.1.3

Factor $\sqrt{6} y$ out of $\sqrt{6} y (3) + \sqrt{6} y (1)$ .

$x = \frac{\frac{\sqrt{6} y (3 + 1)}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\frac{\sqrt{6} y (3 + 1)}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.5.2

Add $3$ and $1$ .

$x = \frac{\frac{\sqrt{6} y \cdot 4}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\frac{\sqrt{6} y \cdot 4}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.6

Move $4$ to the left of $\sqrt{6} y$ .

$x = \frac{\frac{4 \sqrt{6} y}{3}}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.7

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{4 \sqrt{6} y}{3} \cdot \frac{1}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.8

Cancel the common factor of $4$ .

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Step 3.4.4.1.8.1

Factor $4$ out of $4 \sqrt{6} y$ .

$x = \frac{4 (\sqrt{6} y)}{3} \cdot \frac{1}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.8.2

Cancel the common factor.

$x = \frac{4 (\sqrt{6} y)}{3} \cdot \frac{1}{4}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.4.4.1.8.3

Rewrite the expression.

$x = \frac{\sqrt{6} y}{3}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{3}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{3}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{3}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

$x = \frac{\sqrt{6} y}{3}$

$0 = 0$

$z = \frac{\sqrt{3} y}{3}$

Step 3.5

Remove any equations from the system that are always true.

$x = \frac{\sqrt{6} y}{3}, z = \frac{\sqrt{3} y}{3}$