Linear Algebra Examples

$[\begin{array}{c} 1 & 1 & 1 \\ 4 & 10 & 0 \\ 4 \cdot \sqrt{3} & 0 & - 2 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 600 \\ 4500 \\ 0 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 1 & 1 & 1 \\ 4 & 10 & 0 \\ 4 \sqrt{3} & 0 & - 2 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}]$ .

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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.

$[\begin{array}{c} 1 x + 1 y + 1 z \\ 4 x + 10 y + 0 z \\ 4 \sqrt{3} x + 0 y - 2 z \end{array}] = [\begin{array}{c} 600 \\ 4500 \\ 0 \end{array}]$

Step 1.3

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} x + y + z \\ 4 x + 10 y \\ 4 \sqrt{3} x - 2 z \end{array}] = [\begin{array}{c} 600 \\ 4500 \\ 0 \end{array}]$

Step 2

The matrix equation can be written as a set of equations.

$x + y + z = 600$

$4 x + 10 y = 4500$

$4 \sqrt{3} x - 2 z = 0$

Step 3

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

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Step 3.1

Subtract $y$ from both sides of the equation.

$x + z = 600 - y$

$4 x + 10 y = 4500$

$4 \sqrt{3} x - 2 z = 0$

Step 3.2

Subtract $z$ from both sides of the equation.

$x = 600 - y - z$

$4 x + 10 y = 4500$

$4 \sqrt{3} x - 2 z = 0$

$x = 600 - y - z$

$4 x + 10 y = 4500$

$4 \sqrt{3} x - 2 z = 0$

Step 4

Replace all occurrences of $x$ with $600 - y - z$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $4 x + 10 y = 4500$ with $600 - y - z$ .

$4 (600 - y - z) + 10 y = 4500$

$x = 600 - y - z$

$4 \sqrt{3} x - 2 z = 0$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $4 (600 - y - z) + 10 y$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$4 \cdot 600 + 4 (- y) + 4 (- z) + 10 y = 4500$

$x = 600 - y - z$

$4 \sqrt{3} x - 2 z = 0$

Step 4.2.1.1.2

Simplify.

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Step 4.2.1.1.2.1

Multiply $4$ by $600$ .

$2400 + 4 (- y) + 4 (- z) + 10 y = 4500$

$x = 600 - y - z$

$4 \sqrt{3} x - 2 z = 0$

Step 4.2.1.1.2.2

Multiply $- 1$ by $4$ .

$2400 - 4 y + 4 (- z) + 10 y = 4500$

$x = 600 - y - z$

$4 \sqrt{3} x - 2 z = 0$

Step 4.2.1.1.2.3

Multiply $- 1$ by $4$ .

$2400 - 4 y - 4 z + 10 y = 4500$

$x = 600 - y - z$

$4 \sqrt{3} x - 2 z = 0$

$2400 - 4 y - 4 z + 10 y = 4500$

$x = 600 - y - z$

$4 \sqrt{3} x - 2 z = 0$

$2400 - 4 y - 4 z + 10 y = 4500$

$x = 600 - y - z$

$4 \sqrt{3} x - 2 z = 0$

Step 4.2.1.2

Add $- 4 y$ and $10 y$ .

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$4 \sqrt{3} x - 2 z = 0$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$4 \sqrt{3} x - 2 z = 0$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$4 \sqrt{3} x - 2 z = 0$

Step 4.3

Replace all occurrences of $x$ in $4 \sqrt{3} x - 2 z = 0$ with $600 - y - z$ .

$4 \sqrt{3} (600 - y - z) - 2 z = 0$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 4.4

Simplify the left side.

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

Apply the distributive property.

$4 \sqrt{3} \cdot 600 + 4 \sqrt{3} (- y) + 4 \sqrt{3} (- z) - 2 z = 0$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 4.4.1.2

Simplify.

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Step 4.4.1.2.1

Multiply $600$ by $4$ .

$2400 \sqrt{3} + 4 \sqrt{3} (- y) + 4 \sqrt{3} (- z) - 2 z = 0$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 4.4.1.2.2

Multiply $- 1$ by $4$ .

$2400 \sqrt{3} - 4 \sqrt{3} y + 4 \sqrt{3} (- z) - 2 z = 0$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 4.4.1.2.3

Multiply $- 1$ by $4$ .

$2400 \sqrt{3} - 4 \sqrt{3} y - 4 \sqrt{3} z - 2 z = 0$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$2400 \sqrt{3} - 4 \sqrt{3} y - 4 \sqrt{3} z - 2 z = 0$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$2400 \sqrt{3} - 4 \sqrt{3} y - 4 \sqrt{3} z - 2 z = 0$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$2400 \sqrt{3} - 4 \sqrt{3} y - 4 \sqrt{3} z - 2 z = 0$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$2400 \sqrt{3} - 4 \sqrt{3} y - 4 \sqrt{3} z - 2 z = 0$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5

Solve for $y$ in $2400 \sqrt{3} - 4 \sqrt{3} y - 4 \sqrt{3} z - 2 z = 0$ .

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Step 5.1

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

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Step 5.1.1

Subtract $2400 \sqrt{3}$ from both sides of the equation.

$- 4 \sqrt{3} y - 4 \sqrt{3} z - 2 z = - 2400 \sqrt{3}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.1.2

Add $4 \sqrt{3} z$ to both sides of the equation.

$- 4 \sqrt{3} y - 2 z = - 2400 \sqrt{3} + 4 \sqrt{3} z$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.1.3

Add $2 z$ to both sides of the equation.

$- 4 \sqrt{3} y = - 2400 \sqrt{3} + 4 \sqrt{3} z + 2 z$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$- 4 \sqrt{3} y = - 2400 \sqrt{3} + 4 \sqrt{3} z + 2 z$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2

Divide each term in $- 4 \sqrt{3} y = - 2400 \sqrt{3} + 4 \sqrt{3} z + 2 z$ by $- 4 \sqrt{3}$ and simplify.

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Step 5.2.1

Divide each term in $- 4 \sqrt{3} y = - 2400 \sqrt{3} + 4 \sqrt{3} z + 2 z$ by $- 4 \sqrt{3}$ .

$\frac{- 4 \sqrt{3} y}{- 4 \sqrt{3}} = \frac{- 2400 \sqrt{3}}{- 4 \sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 4$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 4 \sqrt{3} y}{- 4 \sqrt{3}} = \frac{- 2400 \sqrt{3}}{- 4 \sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{\sqrt{3} y}{\sqrt{3}} = \frac{- 2400 \sqrt{3}}{- 4 \sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$\frac{\sqrt{3} y}{\sqrt{3}} = \frac{- 2400 \sqrt{3}}{- 4 \sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.2.2

Cancel the common factor of $\sqrt{3}$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{\sqrt{3} y}{\sqrt{3}} = \frac{- 2400 \sqrt{3}}{- 4 \sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 2400 \sqrt{3}}{- 4 \sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = \frac{- 2400 \sqrt{3}}{- 4 \sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = \frac{- 2400 \sqrt{3}}{- 4 \sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Cancel the common factor of $- 2400$ and $- 4$ .

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Step 5.2.3.1.1.1

Factor $- 4$ out of $- 2400 \sqrt{3}$ .

$y = \frac{- 4 (600 \sqrt{3})}{- 4 \sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.1.2

Cancel the common factors.

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Step 5.2.3.1.1.2.1

Factor $- 4$ out of $- 4 \sqrt{3}$ .

$y = \frac{- 4 (600 \sqrt{3})}{- 4 \sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{- 4 (600 \sqrt{3})}{- 4 \sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{600 \sqrt{3}}{\sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = \frac{600 \sqrt{3}}{\sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = \frac{600 \sqrt{3}}{\sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.2

Cancel the common factor of $\sqrt{3}$ .

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Step 5.2.3.1.2.1

Cancel the common factor.

$y = \frac{600 \sqrt{3}}{\sqrt{3}} + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.2.2

Divide $600$ by $1$ .

$y = 600 + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 + \frac{4 \sqrt{3} z}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.3

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

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Step 5.2.3.1.3.1

Factor $4$ out of $4 \sqrt{3} z$ .

$y = 600 + \frac{4 (\sqrt{3} z)}{- 4 \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.3.2

Cancel the common factors.

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Step 5.2.3.1.3.2.1

Factor $4$ out of $- 4 \sqrt{3}$ .

$y = 600 + \frac{4 (\sqrt{3} z)}{4 (- \sqrt{3})} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.3.2.2

Cancel the common factor.

$y = 600 + \frac{4 (\sqrt{3} z)}{4 (- \sqrt{3})} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.3.2.3

Rewrite the expression.

$y = 600 + \frac{\sqrt{3} z}{- \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 + \frac{\sqrt{3} z}{- \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 + \frac{\sqrt{3} z}{- \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.4

Cancel the common factor of $\sqrt{3}$ .

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Step 5.2.3.1.4.1

Cancel the common factor.

$y = 600 + \frac{\sqrt{3} z}{- \sqrt{3}} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.4.2

Rewrite the expression.

$y = 600 + \frac{z}{- 1} + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.4.3

Move the negative one from the denominator of $\frac{z}{- 1}$ .

$y = 600 - 1 \cdot z + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 - 1 \cdot z + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.5

Rewrite $- 1 \cdot z$ as $- z$ .

$y = 600 - z + \frac{2 z}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.6

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

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Step 5.2.3.1.6.1

Factor $2$ out of $2 z$ .

$y = 600 - z + \frac{2 (z)}{- 4 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.6.2

Cancel the common factors.

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Step 5.2.3.1.6.2.1

Factor $2$ out of $- 4 \sqrt{3}$ .

$y = 600 - z + \frac{2 (z)}{2 (- 2 \sqrt{3})}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.6.2.2

Cancel the common factor.

$y = 600 - z + \frac{2 z}{2 (- 2 \sqrt{3})}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.6.2.3

Rewrite the expression.

$y = 600 - z + \frac{z}{- 2 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 - z + \frac{z}{- 2 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 - z + \frac{z}{- 2 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.7

Move the negative in front of the fraction.

$y = 600 - z - \frac{z}{2 \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.8

Multiply $\frac{z}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = 600 - z - (\frac{z}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.9

Combine and simplify the denominator.

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Step 5.2.3.1.9.1

Multiply $\frac{z}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = 600 - z - \frac{z \sqrt{3}}{2 \sqrt{3} \sqrt{3}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.9.2

Move $\sqrt{3}$ .

$y = 600 - z - \frac{z \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.9.3

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

$y = 600 - z - \frac{z \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.9.4

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

$y = 600 - z - \frac{z \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.9.5

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

$y = 600 - z - \frac{z \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.9.6

Add $1$ and $1$ .

$y = 600 - z - \frac{z \sqrt{3}}{2 {\sqrt{3}}^{2}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.9.7

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

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Step 5.2.3.1.9.7.1

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

$y = 600 - z - \frac{z \sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.9.7.2

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

$y = 600 - z - \frac{z \sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.9.7.3

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

$y = 600 - z - \frac{z \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.9.7.4

Cancel the common factor of $2$ .

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Step 5.2.3.1.9.7.4.1

Cancel the common factor.

$y = 600 - z - \frac{z \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.9.7.4.2

Rewrite the expression.

$y = 600 - z - \frac{z \sqrt{3}}{2 \cdot 3}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 - z - \frac{z \sqrt{3}}{2 \cdot 3}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.9.7.5

Evaluate the exponent.

$y = 600 - z - \frac{z \sqrt{3}}{2 \cdot 3}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 - z - \frac{z \sqrt{3}}{2 \cdot 3}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 - z - \frac{z \sqrt{3}}{2 \cdot 3}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 5.2.3.1.10

Multiply $2$ by $3$ .

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$2400 + 6 y - 4 z = 4500$

$x = 600 - y - z$

Step 6

Replace all occurrences of $y$ with $600 - z - \frac{z \sqrt{3}}{6}$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $2400 + 6 y - 4 z = 4500$ with $600 - z - \frac{z \sqrt{3}}{6}$ .

$2400 + 6 (600 - z - \frac{z \sqrt{3}}{6}) - 4 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify $2400 + 6 (600 - z - \frac{z \sqrt{3}}{6}) - 4 z$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Apply the distributive property.

$2400 + 6 \cdot 600 + 6 (- z) + 6 (- \frac{z \sqrt{3}}{6}) - 4 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

Step 6.2.1.1.2

Simplify.

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Step 6.2.1.1.2.1

Multiply $6$ by $600$ .

$2400 + 3600 + 6 (- z) + 6 (- \frac{z \sqrt{3}}{6}) - 4 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

Step 6.2.1.1.2.2

Multiply $- 1$ by $6$ .

$2400 + 3600 - 6 z + 6 (- \frac{z \sqrt{3}}{6}) - 4 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

Step 6.2.1.1.2.3

Cancel the common factor of $6$ .

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Step 6.2.1.1.2.3.1

Move the leading negative in $- \frac{z \sqrt{3}}{6}$ into the numerator.

$2400 + 3600 - 6 z + 6 (\frac{- z \sqrt{3}}{6}) - 4 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

Step 6.2.1.1.2.3.2

Cancel the common factor.

$2400 + 3600 - 6 z + 6 (\frac{- z \sqrt{3}}{6}) - 4 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

Step 6.2.1.1.2.3.3

Rewrite the expression.

$2400 + 3600 - 6 z - z \sqrt{3} - 4 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

$2400 + 3600 - 6 z - z \sqrt{3} - 4 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

$2400 + 3600 - 6 z - z \sqrt{3} - 4 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

$2400 + 3600 - 6 z - z \sqrt{3} - 4 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

Step 6.2.1.2

Simplify by adding terms.

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Step 6.2.1.2.1

Add $2400$ and $3600$ .

$6000 - 6 z - z \sqrt{3} - 4 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

Step 6.2.1.2.2

Subtract $4 z$ from $- 6 z$ .

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - y - z$

Step 6.3

Replace all occurrences of $y$ in $x = 600 - y - z$ with $600 - z - \frac{z \sqrt{3}}{6}$ .

$x = 600 - (600 - z - \frac{z \sqrt{3}}{6}) - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $600 - (600 - z - \frac{z \sqrt{3}}{6}) - z$ .

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Step 6.4.1.1

Simplify each term.

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Step 6.4.1.1.1

Apply the distributive property.

$x = 600 - 1 \cdot 600 + z + \frac{z \sqrt{3}}{6} - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 6.4.1.1.2

Simplify.

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Step 6.4.1.1.2.1

Multiply $- 1$ by $600$ .

$x = 600 - 600 + z + \frac{z \sqrt{3}}{6} - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 6.4.1.1.2.2

Multiply $- - z$ .

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Step 6.4.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$x = 600 - 600 + 1 z + \frac{z \sqrt{3}}{6} - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 6.4.1.1.2.2.2

Multiply $z$ by $1$ .

$x = 600 - 600 + z + \frac{z \sqrt{3}}{6} - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - 600 + z + \frac{z \sqrt{3}}{6} - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 6.4.1.1.2.3

Multiply $- - \frac{z \sqrt{3}}{6}$ .

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Step 6.4.1.1.2.3.1

Multiply $- 1$ by $- 1$ .

$x = 600 - 600 + z + 1 (\frac{z \sqrt{3}}{6}) - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 6.4.1.1.2.3.2

Multiply $\frac{z \sqrt{3}}{6}$ by $1$ .

$x = 600 - 600 + z + \frac{z \sqrt{3}}{6} - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - 600 + z + \frac{z \sqrt{3}}{6} - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - 600 + z + \frac{z \sqrt{3}}{6} - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = 600 - 600 + z + \frac{z \sqrt{3}}{6} - z$

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 6.4.1.2

Combine the opposite terms in $600 - 600 + z + \frac{z \sqrt{3}}{6} - z$ .

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Step 6.4.1.2.1

Subtract $600$ from $600$ .

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

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 6.4.1.2.2

Subtract $z$ from $z$ .

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

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 6.4.1.2.3

Add $0$ and $\frac{z \sqrt{3}}{6}$ .

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

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 6.4.1.2.4

Add $\frac{z \sqrt{3}}{6}$ and $0$ .

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

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

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

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

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

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

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

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

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

$6000 - z \sqrt{3} - 10 z = 4500$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7

Solve for $z$ in $6000 - z \sqrt{3} - 10 z = 4500$ .

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Step 7.1

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

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Step 7.1.1

Subtract $6000$ from both sides of the equation.

$- z \sqrt{3} - 10 z = 4500 - 6000$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.1.2

Subtract $6000$ from $4500$ .

$- z \sqrt{3} - 10 z = - 1500$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$- z \sqrt{3} - 10 z = - 1500$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.2

Factor $- z$ out of $- z \sqrt{3} - 10 z$ .

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Step 7.2.1

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

$- z \sqrt{3} - 10 z = - 1500$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.2.2

Factor $- z$ out of $- 10 z$ .

$- z \sqrt{3} - z \cdot 10 = - 1500$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.2.3

Factor $- z$ out of $- z (\sqrt{3}) - z (10)$ .

$- z (\sqrt{3} + 10) = - 1500$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$- z (\sqrt{3} + 10) = - 1500$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3

Divide each term in $- z (\sqrt{3} + 10) = - 1500$ by $- \sqrt{3} - 10$ and simplify.

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Step 7.3.1

Divide each term in $- z (\sqrt{3} + 10) = - 1500$ by $- \sqrt{3} - 10$ .

$\frac{- z (\sqrt{3} + 10)}{- \sqrt{3} - 10} = \frac{- 1500}{- \sqrt{3} - 10}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $\sqrt{3} + 10$ and $- \sqrt{3} - 10$ .

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Step 7.3.2.1.1

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

$\frac{- z (- 1 (- \sqrt{3}) + 10)}{- \sqrt{3} - 10} = \frac{- 1500}{- \sqrt{3} - 10}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.2.1.2

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{- z (- 1 (- \sqrt{3}) - 1 \cdot - 10)}{- \sqrt{3} - 10} = \frac{- 1500}{- \sqrt{3} - 10}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.2.1.3

Factor $- 1$ out of $- 1 (- \sqrt{3}) - 1 (- 10)$ .

$\frac{- z (- 1 (- \sqrt{3} - 10))}{- \sqrt{3} - 10} = \frac{- 1500}{- \sqrt{3} - 10}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.2.1.4

Cancel the common factor.

$\frac{- z (- 1 (- \sqrt{3} - 10))}{- \sqrt{3} - 10} = \frac{- 1500}{- \sqrt{3} - 10}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.2.1.5

Divide $- z \cdot (- 1)$ by $1$ .

$- z \cdot - 1 = \frac{- 1500}{- \sqrt{3} - 10}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$- z \cdot - 1 = \frac{- 1500}{- \sqrt{3} - 10}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.2.2

Multiply $- z (- 1)$ .

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Step 7.3.2.2.1

Multiply $- 1$ by $- 1$ .

$1 z = \frac{- 1500}{- \sqrt{3} - 10}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.2.2.2

Multiply $z$ by $1$ .

$z = \frac{- 1500}{- \sqrt{3} - 10}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$z = \frac{- 1500}{- \sqrt{3} - 10}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$z = \frac{- 1500}{- \sqrt{3} - 10}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Move the negative in front of the fraction.

$z = - \frac{1500}{- \sqrt{3} - 10}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3.2

Multiply $\frac{1500}{- \sqrt{3} - 10}$ by $\frac{- \sqrt{3} + 10}{- \sqrt{3} + 10}$ .

$z = - (\frac{1500}{- \sqrt{3} - 10} \cdot \frac{- \sqrt{3} + 10}{- \sqrt{3} + 10})$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3.3

Multiply $\frac{1500}{- \sqrt{3} - 10}$ by $\frac{- \sqrt{3} + 10}{- \sqrt{3} + 10}$ .

$z = - \frac{1500 (- \sqrt{3} + 10)}{(- \sqrt{3} - 10) (- \sqrt{3} + 10)}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3.4

Expand the denominator using the FOIL method.

$z = - \frac{1500 (- \sqrt{3} + 10)}{{\sqrt{3}}^{2} - 10 \sqrt{3} + 10 \sqrt{3} - 100}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3.5

Simplify.

$z = - \frac{1500 (- \sqrt{3} + 10)}{- 97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3.6

Move the negative in front of the fraction.

$z = \frac{1500 (- \sqrt{3} + 10)}{97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3.7

Multiply $- - \frac{1500 (- \sqrt{3} + 10)}{97}$ .

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Step 7.3.3.7.1

Multiply $- 1$ by $- 1$ .

$z = 1 (\frac{1500 (- \sqrt{3} + 10)}{97})$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3.7.2

Multiply $\frac{1500 (- \sqrt{3} + 10)}{97}$ by $1$ .

$z = \frac{1500 (- \sqrt{3} + 10)}{97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$z = \frac{1500 (- \sqrt{3} + 10)}{97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3.8

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

$z = \frac{1500 (- (\sqrt{3}) + 10)}{97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3.9

Rewrite $10$ as $- 1 (- 10)$ .

$z = \frac{1500 (- (\sqrt{3}) - 1 \cdot - 10)}{97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3.10

Factor $- 1$ out of $- (\sqrt{3}) - 1 (- 10)$ .

$z = \frac{1500 (- (\sqrt{3} - 10))}{97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3.11

Simplify the expression.

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Step 7.3.3.11.1

Rewrite $- (\sqrt{3} - 10)$ as $- 1 (\sqrt{3} - 10)$ .

$z = \frac{1500 (- 1 (\sqrt{3} - 10))}{97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 7.3.3.11.2

Move the negative in front of the fraction.

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

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

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8

Replace all occurrences of $z$ with $- \frac{1500 (\sqrt{3} - 10)}{97}$ in each equation.

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Step 8.1

Replace all occurrences of $z$ in $x = \frac{z \sqrt{3}}{6}$ with $- \frac{1500 (\sqrt{3} - 10)}{97}$ .

$x = \frac{(- \frac{1500 (\sqrt{3} - 10)}{97}) \sqrt{3}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{(- \frac{1500 (\sqrt{3} - 10)}{97}) \sqrt{3}}{6}$ .

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Step 8.2.1.1

Combine $\sqrt{3}$ and $\frac{1500 (\sqrt{3} - 10)}{97}$ .

$x = \frac{- \frac{\sqrt{3} (1500 (\sqrt{3} - 10))}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.2

Simplify the numerator.

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Step 8.2.1.2.1

Group $\sqrt{3} - 10$ and $\sqrt{3}$ together.

$x = \frac{- \frac{(\sqrt{3} - 10) \sqrt{3} \cdot 1500}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.2.2

Apply the distributive property.

$x = \frac{- \frac{(\sqrt{3} \sqrt{3} - 10 \sqrt{3}) \cdot 1500}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.2.3

Combine using the product rule for radicals.

$x = \frac{- \frac{(\sqrt{3 \cdot 3} - 10 \sqrt{3}) \cdot 1500}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.2.4

Simplify each term.

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Step 8.2.1.2.4.1

Multiply $3$ by $3$ .

$x = \frac{- \frac{(\sqrt{9} - 10 \sqrt{3}) \cdot 1500}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.2.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{- \frac{(\sqrt{3^{2}} - 10 \sqrt{3}) \cdot 1500}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.2.4.3

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- \frac{(3 - 10 \sqrt{3}) \cdot 1500}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = \frac{- \frac{(3 - 10 \sqrt{3}) \cdot 1500}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.2.5

Move $1500$ to the left of $3 - 10 \sqrt{3}$ .

$x = \frac{- \frac{1500 (3 - 10 \sqrt{3})}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = \frac{- \frac{1500 (3 - 10 \sqrt{3})}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.3

Simplify the numerator.

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Step 8.2.1.3.1

Apply the distributive property.

$x = \frac{- \frac{1500 \cdot 3 + 1500 (- 10 \sqrt{3})}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.3.2

Multiply $1500$ by $3$ .

$x = \frac{- \frac{4500 + 1500 (- 10 \sqrt{3})}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.3.3

Multiply $- 10$ by $1500$ .

$x = \frac{- \frac{4500 - 15000 \sqrt{3}}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = \frac{- \frac{4500 - 15000 \sqrt{3}}{97}}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{4500 - 15000 \sqrt{3}}{97} \cdot \frac{1}{6}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.5

Multiply $- \frac{4500 - 15000 \sqrt{3}}{97} \cdot \frac{1}{6}$ .

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Step 8.2.1.5.1

Multiply $\frac{1}{6}$ by $\frac{4500 - 15000 \sqrt{3}}{97}$ .

$x = - \frac{4500 - 15000 \sqrt{3}}{6 \cdot 97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.5.2

Multiply $6$ by $97$ .

$x = - \frac{4500 - 15000 \sqrt{3}}{582}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = - \frac{4500 - 15000 \sqrt{3}}{582}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.6

Cancel the common factor of $4500 - 15000 \sqrt{3}$ and $582$ .

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Step 8.2.1.6.1

Factor $6$ out of $4500$ .

$x = - \frac{6 (750) - 15000 \sqrt{3}}{582}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.6.2

Factor $6$ out of $- 15000 \sqrt{3}$ .

$x = - \frac{6 (750) + 6 (- 2500 \sqrt{3})}{582}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.6.3

Factor $6$ out of $6 (750) + 6 (- 2500 \sqrt{3})$ .

$x = - \frac{6 (750 - 2500 \sqrt{3})}{582}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.6.4

Cancel the common factors.

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Step 8.2.1.6.4.1

Factor $6$ out of $582$ .

$x = - \frac{6 (750 - 2500 \sqrt{3})}{6 \cdot 97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.6.4.2

Cancel the common factor.

$x = - \frac{6 (750 - 2500 \sqrt{3})}{6 \cdot 97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.2.1.6.4.3

Rewrite the expression.

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 - z - \frac{z \sqrt{3}}{6}$

Step 8.3

Replace all occurrences of $z$ in $y = 600 - z - \frac{z \sqrt{3}}{6}$ with $- \frac{1500 (\sqrt{3} - 10)}{97}$ .

$y = 600 - (- \frac{1500 (\sqrt{3} - 10)}{97}) - \frac{(- \frac{1500 (\sqrt{3} - 10)}{97}) \sqrt{3}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4

Simplify the right side.

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Step 8.4.1

Simplify $600 - (- \frac{1500 (\sqrt{3} - 10)}{97}) - \frac{(- \frac{1500 (\sqrt{3} - 10)}{97}) \sqrt{3}}{6}$ .

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Step 8.4.1.1

Simplify each term.

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Step 8.4.1.1.1

Multiply $- (- \frac{1500 (\sqrt{3} - 10)}{97})$ .

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Step 8.4.1.1.1.1

Multiply $- 1$ by $- 1$ .

$y = 600 + 1 (\frac{1500 (\sqrt{3} - 10)}{97}) - \frac{(- \frac{1500 (\sqrt{3} - 10)}{97}) \sqrt{3}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.1.2

Multiply $\frac{1500 (\sqrt{3} - 10)}{97}$ by $1$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{(- \frac{1500 (\sqrt{3} - 10)}{97}) \sqrt{3}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{(- \frac{1500 (\sqrt{3} - 10)}{97}) \sqrt{3}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.2

Combine $\sqrt{3}$ and $\frac{1500 (\sqrt{3} - 10)}{97}$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{\sqrt{3} (1500 (\sqrt{3} - 10))}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.3

Simplify the numerator.

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Step 8.4.1.1.3.1

Group $\sqrt{3} - 10$ and $\sqrt{3}$ together.

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{(\sqrt{3} - 10) \sqrt{3} \cdot 1500}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.3.2

Apply the distributive property.

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{(\sqrt{3} \sqrt{3} - 10 \sqrt{3}) \cdot 1500}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.3.3

Combine using the product rule for radicals.

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{(\sqrt{3 \cdot 3} - 10 \sqrt{3}) \cdot 1500}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.3.4

Simplify each term.

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Step 8.4.1.1.3.4.1

Multiply $3$ by $3$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{(\sqrt{9} - 10 \sqrt{3}) \cdot 1500}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.3.4.2

Rewrite $9$ as $3^{2}$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{(\sqrt{3^{2}} - 10 \sqrt{3}) \cdot 1500}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.3.4.3

Pull terms out from under the radical, assuming positive real numbers.

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{(3 - 10 \sqrt{3}) \cdot 1500}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{(3 - 10 \sqrt{3}) \cdot 1500}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.3.5

Move $1500$ to the left of $3 - 10 \sqrt{3}$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{1500 (3 - 10 \sqrt{3})}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{1500 (3 - 10 \sqrt{3})}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.4

Simplify the numerator.

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Step 8.4.1.1.4.1

Apply the distributive property.

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{1500 \cdot 3 + 1500 (- 10 \sqrt{3})}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.4.2

Multiply $1500$ by $3$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{4500 + 1500 (- 10 \sqrt{3})}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.4.3

Multiply $- 10$ by $1500$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{4500 - 15000 \sqrt{3}}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - \frac{- \frac{4500 - 15000 \sqrt{3}}{97}}{6}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.5

Multiply the numerator by the reciprocal of the denominator.

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} - (- \frac{4500 - 15000 \sqrt{3}}{97} \cdot \frac{1}{6})$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.6

Multiply $- \frac{4500 - 15000 \sqrt{3}}{97} \cdot \frac{1}{6}$ .

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Step 8.4.1.1.6.1

Multiply $\frac{1}{6}$ by $\frac{4500 - 15000 \sqrt{3}}{97}$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{4500 - 15000 \sqrt{3}}{6 \cdot 97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.6.2

Multiply $6$ by $97$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{4500 - 15000 \sqrt{3}}{582}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{4500 - 15000 \sqrt{3}}{582}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.7

Cancel the common factor of $4500 - 15000 \sqrt{3}$ and $582$ .

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Step 8.4.1.1.7.1

Factor $6$ out of $4500$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{6 (750) - 15000 \sqrt{3}}{582}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.7.2

Factor $6$ out of $- 15000 \sqrt{3}$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{6 (750) + 6 (- 2500 \sqrt{3})}{582}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.7.3

Factor $6$ out of $6 (750) + 6 (- 2500 \sqrt{3})$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{6 (750 - 2500 \sqrt{3})}{582}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.7.4

Cancel the common factors.

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Step 8.4.1.1.7.4.1

Factor $6$ out of $582$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{6 (750 - 2500 \sqrt{3})}{6 \cdot 97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.7.4.2

Cancel the common factor.

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{6 (750 - 2500 \sqrt{3})}{6 \cdot 97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.7.4.3

Rewrite the expression.

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{750 - 2500 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{750 - 2500 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{750 - 2500 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.8

Multiply $- - \frac{750 - 2500 \sqrt{3}}{97}$ .

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Step 8.4.1.1.8.1

Multiply $- 1$ by $- 1$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + 1 (\frac{750 - 2500 \sqrt{3}}{97})$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.1.8.2

Multiply $\frac{750 - 2500 \sqrt{3}}{97}$ by $1$ .

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{750 - 2500 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{750 - 2500 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 + \frac{1500 (\sqrt{3} - 10)}{97} + \frac{750 - 2500 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.2

Combine the numerators over the common denominator.

$y = 600 + \frac{1500 (\sqrt{3} - 10) + 750 - 2500 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.3

Simplify each term.

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Step 8.4.1.3.1

Apply the distributive property.

$y = 600 + \frac{1500 \sqrt{3} + 1500 \cdot - 10 + 750 - 2500 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.3.2

Multiply $1500$ by $- 10$ .

$y = 600 + \frac{1500 \sqrt{3} - 15000 + 750 - 2500 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 + \frac{1500 \sqrt{3} - 15000 + 750 - 2500 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.4

Simplify by adding terms.

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Step 8.4.1.4.1

Subtract $2500 \sqrt{3}$ from $1500 \sqrt{3}$ .

$y = 600 + \frac{- 15000 + 750 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.4.2

Add $- 15000$ and $750$ .

$y = 600 + \frac{- 14250 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = 600 + \frac{- 14250 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.5

To write $600$ as a fraction with a common denominator, multiply by $\frac{97}{97}$ .

$y = 600 \cdot \frac{97}{97} + \frac{- 14250 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.6

Combine fractions.

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Step 8.4.1.6.1

Combine $600$ and $\frac{97}{97}$ .

$y = \frac{600 \cdot 97}{97} + \frac{- 14250 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.6.2

Combine the numerators over the common denominator.

$y = \frac{600 \cdot 97 - 14250 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = \frac{600 \cdot 97 - 14250 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.7

Simplify the numerator.

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Step 8.4.1.7.1

Multiply $600$ by $97$ .

$y = \frac{58200 - 14250 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 8.4.1.7.2

Subtract $14250$ from $58200$ .

$y = \frac{43950 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = \frac{43950 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = \frac{43950 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = \frac{43950 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

$y = \frac{43950 - 1000 \sqrt{3}}{97}$

$x = - \frac{750 - 2500 \sqrt{3}}{97}$

$z = - \frac{1500 (\sqrt{3} - 10)}{97}$

Step 9

List all of the solutions.

$y = \frac{43950 - 1000 \sqrt{3}}{97}, x = - \frac{750 - 2500 \sqrt{3}}{97}, z = - \frac{1500 \sqrt{3} - 15000}{97}$