Precalculus Examples

$x y = z^{2}$ , $x + y + z = 28$ , $x^{2} + y^{2} + z^{2} = 2128$

Step 1

Choose two equations and eliminate one variable. In this case, eliminate $z^{2}$ .

$x y = z^{2}$

$x^{2} + y^{2} + z^{2} = 2128$

Step 2

Eliminate $z^{2}$ from the system.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Reorder $x^{2}$ and $y^{2}$ .

$y^{2} + x^{2} + z^{2} = 2128$

$x y = z^{2}$

$y^{2} + x^{2} + z^{2} = 2128$

$x y = z^{2}$

Step 2.2

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

$x y - z^{2} = 0, y^{2} + x^{2} + z^{2} = 2128$

Step 2.3

Add the two equations together to eliminate $z^{2}$ from the system.

		$x$	$y$	$-$	$z^{2}$	$=$				$0$
$+$				$+$	$z^{2}$	$=$	$2$	$1$	$2$	$8$
$x^{2}$	$+$	$x$	$y$	$+$		$=$	$2$	$1$	$2$	$8$

Step 2.4

The resultant equation has $z^{2}$ eliminated.

$x y + y^{2} + x^{2} = 2128$

Step 3

Take the resultant equation and the third original equation and eliminate another variable. In this case, eliminate $x$ .

$x y + y^{2} + x^{2} = 2128$

$x + y + z = 28$

Step 4

Eliminate $x$ from the system.

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Step 4.1

Multiply each equation by the value that makes the coefficients of $x$ opposite.

$x y + y^{2} + x^{2} = 2128$

$(- y) \cdot (x + y + z) = (- y) (28)$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $(- y) \cdot (x + y + z)$ .

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Step 4.2.1.1.1

Apply the distributive property.

$x y + y^{2} + x^{2} = 2128$

$- y x - y \cdot y - y z = (- y) (28)$

Step 4.2.1.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 4.2.1.1.2.1

Move $y$ .

$x y + y^{2} + x^{2} = 2128$

$- y x - (y \cdot y) - y z = (- y) (28)$

Step 4.2.1.1.2.2

Multiply $y$ by $y$ .

$x y + y^{2} + x^{2} = 2128$

$- y x - y^{2} - y z = (- y) (28)$

$x y + y^{2} + x^{2} = 2128$

$- y x - y^{2} - y z = (- y) (28)$

Step 4.2.1.1.3

Move $y$ .

$x y + y^{2} + x^{2} = 2128$

$- 1 x y - y^{2} - y z = (- y) (28)$

Step 4.2.1.1.4

Move $- y^{2}$ .

$x y + y^{2} + x^{2} = 2128$

$- 1 x y - y z - y^{2} = (- y) (28)$

$x y + y^{2} + x^{2} = 2128$

$- 1 x y - y z - y^{2} = (- y) (28)$

$x y + y^{2} + x^{2} = 2128$

$- 1 x y - y z - y^{2} = (- y) (28)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $28$ by $- 1$ .

$x y + y^{2} + x^{2} = 2128$

$- 1 x y - y z - y^{2} = - 28 y$

$x y + y^{2} + x^{2} = 2128$

$- 1 x y - y z - y^{2} = - 28 y$

Step 4.2.3

Reorder the polynomial.

$x y + y^{2} + x^{2} = 2128$

$- y^{2} - 1 x y - y z = - 28 y$

$x y + y^{2} + x^{2} = 2128$

$- y^{2} - 1 x y - y z = - 28 y$

Step 4.3

Add the two equations together to eliminate $x$ from the system.

$x$

$y$

$+$

$y^{2}$

$+$

$x^{2}$

$=$

$2$

$1$

$2$

$8$

$+$

$-$

$1$

$x$

$y$

$=$

$-$

$y$

$^$

$-$

$y$

$z$

$y^{2}$

$+$

$x^{2}$

$=$

$-$

$2$

$8$

$y$

Step 4.4

The resultant equation has $x$ eliminated.

$y^{2} + x^{2} - y^{2} - y z = 2128 - 28 y$

Step 4.5

Move all terms not containing $y^{2}$ to the right side of the equation.

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Step 4.5.1

Subtract $x^{2}$ from both sides of the equation.

$y^{2} - y^{2} - y z = 2128 - 28 y - x^{2}$

Step 4.5.2

Add $y^{2}$ to both sides of the equation.

$y^{2} - y z = 2128 - 28 y - x^{2} + y^{2}$

Step 4.5.3

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

$y^{2} = 2128 - 28 y - x^{2} + y^{2} + y z$

Step 5

Substitute the value of $y^{2}$ into an equation with $z^{2}$ eliminated already and solve for the remaining variable.

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Step 5.1

Substitute the value of $y^{2}$ into an equation with $z^{2}$ eliminated already.

$x y + 2128 - 28 y - x^{2} + y^{2} + y z + x^{2} = 2128$

Step 5.2

Solve for $x$ .

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Step 5.2.1

Combine the opposite terms in $x y + 2128 - 28 y - x^{2} + y^{2} + y z + x^{2}$ .

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Step 5.2.1.1

Add $- x^{2}$ and $x^{2}$ .

$x y + 2128 - 28 y + y^{2} + y z + 0 = 2128$

Step 5.2.1.2

Add $x y + 2128 - 28 y + y^{2} + y z$ and $0$ .

$x y + 2128 - 28 y + y^{2} + y z = 2128$

Step 5.2.2

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

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Step 5.2.2.1

Subtract $2128$ from both sides of the equation.

$x y - 28 y + y^{2} + y z = 2128 - 2128$

Step 5.2.2.2

Add $28 y$ to both sides of the equation.

$x y + y^{2} + y z = 2128 - 2128 + 28 y$

Step 5.2.2.3

Subtract $y^{2}$ from both sides of the equation.

$x y + y z = 2128 - 2128 + 28 y - y^{2}$

Step 5.2.2.4

Subtract $y z$ from both sides of the equation.

$x y = 2128 - 2128 + 28 y - y^{2} - y z$

Step 5.2.2.5

Subtract $2128$ from $2128$ .

$x y = 0 + 28 y - y^{2} - y z$

Step 5.2.2.6

Add $0$ and $28 y$ .

$x y = 28 y - y^{2} - y z$

Step 5.2.3

Divide each term in $x y = 28 y - y^{2} - y z$ by $y$ and simplify.

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Step 5.2.3.1

Divide each term in $x y = 28 y - y^{2} - y z$ by $y$ .

$\frac{x y}{y} = \frac{28 y}{y} + \frac{- y^{2}}{y} + \frac{- y z}{y}$

Step 5.2.3.2

Simplify the left side.

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Step 5.2.3.2.1

Cancel the common factor of $y$ .

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Step 5.2.3.2.1.1

Cancel the common factor.

$\frac{x y}{y} = \frac{28 y}{y} + \frac{- y^{2}}{y} + \frac{- y z}{y}$

Step 5.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{28 y}{y} + \frac{- y^{2}}{y} + \frac{- y z}{y}$

Step 5.2.3.3

Simplify the right side.

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Step 5.2.3.3.1

Simplify each term.

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Step 5.2.3.3.1.1

Cancel the common factor of $y$ .

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Step 5.2.3.3.1.1.1

Cancel the common factor.

$x = \frac{28 y}{y} + \frac{- y^{2}}{y} + \frac{- y z}{y}$

Step 5.2.3.3.1.1.2

Divide $28$ by $1$ .

$x = 28 + \frac{- y^{2}}{y} + \frac{- y z}{y}$

Step 5.2.3.3.1.2

Cancel the common factor of $y^{2}$ and $y$ .

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Step 5.2.3.3.1.2.1

Factor $y$ out of $- y^{2}$ .

$x = 28 + \frac{y (- y)}{y} + \frac{- y z}{y}$

Step 5.2.3.3.1.2.2

Cancel the common factors.

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Step 5.2.3.3.1.2.2.1

Raise $y$ to the power of $1$ .

$x = 28 + \frac{y (- y)}{y^{1}} + \frac{- y z}{y}$

Step 5.2.3.3.1.2.2.2

Factor $y$ out of $y^{1}$ .

$x = 28 + \frac{y (- y)}{y \cdot 1} + \frac{- y z}{y}$

Step 5.2.3.3.1.2.2.3

Cancel the common factor.

$x = 28 + \frac{y (- y)}{y \cdot 1} + \frac{- y z}{y}$

Step 5.2.3.3.1.2.2.4

Rewrite the expression.

$x = 28 + \frac{- y}{1} + \frac{- y z}{y}$

Step 5.2.3.3.1.2.2.5

Divide $- y$ by $1$ .

$x = 28 - y + \frac{- y z}{y}$

Step 5.2.3.3.1.3

Cancel the common factor of $y$ .

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Step 5.2.3.3.1.3.1

Cancel the common factor.

$x = 28 - y + \frac{- y z}{y}$

Step 5.2.3.3.1.3.2

Divide $- 1 z$ by $1$ .

$x = 28 - y - 1 z$

Step 5.2.3.3.1.4

Rewrite $- 1 z$ as $- z$ .

$x = 28 - y - z$

Step 6

Substitute the value of each known variable into one of the initial equations and solve for the last variable.

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Step 6.1

Substitute the value of each known variable into one of the initial equations.

$(28 - y - z) y = z^{2}$

Step 6.2

Solve for $y$ .

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Step 6.2.1

Simplify $(28 - y - z) y$ .

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Step 6.2.1.1

Apply the distributive property.

$28 y - y \cdot y - z y = z^{2}$

Step 6.2.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 6.2.1.2.1

Move $y$ .

$28 y - (y \cdot y) - z y = z^{2}$

Step 6.2.1.2.2

Multiply $y$ by $y$ .

$28 y - y^{2} - z y = z^{2}$

Step 6.2.2

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

$28 y - y^{2} - z y - z^{2} = 0$

Step 6.2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.2.4

Substitute the values $a = - 1$ , $b = 28 - z$ , and $c = - z^{2}$ into the quadratic formula and solve for $y$ .

$\frac{- (28 - z) \pm \sqrt{{(28 - z)}^{2} - 4 \cdot (- 1 \cdot (- z^{2}))}}{2 \cdot - 1}$

Step 6.2.5

Simplify.

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Step 6.2.5.1

Simplify the numerator.

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Step 6.2.5.1.1

Apply the distributive property.

$y = \frac{- 1 \cdot 28 + z \pm \sqrt{{(28 - z)}^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.2

Multiply $- 1$ by $28$ .

$y = \frac{- 28 + z \pm \sqrt{{(28 - z)}^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.3

Multiply $- - z$ .

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Step 6.2.5.1.3.1

Multiply $- 1$ by $- 1$ .

$y = \frac{- 28 + 1 z \pm \sqrt{{(28 - z)}^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.3.2

Multiply $z$ by $1$ .

$y = \frac{- 28 + z \pm \sqrt{{(28 - z)}^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.4

Rewrite ${(28 - z)}^{2}$ as $(28 - z) (28 - z)$ .

$y = \frac{- 28 + z \pm \sqrt{(28 - z) (28 - z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.5

Expand $(28 - z) (28 - z)$ using the FOIL Method.

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Step 6.2.5.1.5.1

Apply the distributive property.

$y = \frac{- 28 + z \pm \sqrt{28 (28 - z) - z (28 - z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.5.2

Apply the distributive property.

$y = \frac{- 28 + z \pm \sqrt{28 \cdot 28 + 28 (- z) - z (28 - z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.5.3

Apply the distributive property.

$y = \frac{- 28 + z \pm \sqrt{28 \cdot 28 + 28 (- z) - z \cdot 28 - z (- z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.6

Simplify and combine like terms.

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Step 6.2.5.1.6.1

Simplify each term.

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Step 6.2.5.1.6.1.1

Multiply $28$ by $28$ .

$y = \frac{- 28 + z \pm \sqrt{784 + 28 (- z) - z \cdot 28 - z (- z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.6.1.2

Multiply $- 1$ by $28$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - z \cdot 28 - z (- z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.6.1.3

Multiply $28$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z - z (- z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.6.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z - 1 \cdot (- 1 z \cdot z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.6.1.5

Multiply $z$ by $z$ by adding the exponents.

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Step 6.2.5.1.6.1.5.1

Move $z$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z - 1 \cdot (- 1 (z \cdot z)) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.6.1.5.2

Multiply $z$ by $z$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z - 1 \cdot (- 1 z^{2}) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.6.1.6

Multiply $- 1$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z + 1 z^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.6.1.7

Multiply $z^{2}$ by $1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z + z^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.6.2

Subtract $28 z$ from $- 28 z$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.7

Multiply $- 4 \cdot - 1 \cdot - 1$ .

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Step 6.2.5.1.7.1

Multiply $- 4$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} + 4 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.5.1.7.2

Multiply $4$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2 \cdot - 1}$

Step 6.2.5.2

Multiply $2$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{- 2}$

Step 6.2.5.3

Simplify $\frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{- 2}$ .

$y = \frac{28 - z \pm \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$

Step 6.2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.2.6.1

Simplify the numerator.

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Step 6.2.6.1.1

Apply the distributive property.

$y = \frac{- 1 \cdot 28 + z \pm \sqrt{{(28 - z)}^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.2

Multiply $- 1$ by $28$ .

$y = \frac{- 28 + z \pm \sqrt{{(28 - z)}^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.3

Multiply $- - z$ .

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Step 6.2.6.1.3.1

Multiply $- 1$ by $- 1$ .

$y = \frac{- 28 + 1 z \pm \sqrt{{(28 - z)}^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.3.2

Multiply $z$ by $1$ .

$y = \frac{- 28 + z \pm \sqrt{{(28 - z)}^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.4

Rewrite ${(28 - z)}^{2}$ as $(28 - z) (28 - z)$ .

$y = \frac{- 28 + z \pm \sqrt{(28 - z) (28 - z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.5

Expand $(28 - z) (28 - z)$ using the FOIL Method.

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Step 6.2.6.1.5.1

Apply the distributive property.

$y = \frac{- 28 + z \pm \sqrt{28 (28 - z) - z (28 - z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.5.2

Apply the distributive property.

$y = \frac{- 28 + z \pm \sqrt{28 \cdot 28 + 28 (- z) - z (28 - z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.5.3

Apply the distributive property.

$y = \frac{- 28 + z \pm \sqrt{28 \cdot 28 + 28 (- z) - z \cdot 28 - z (- z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.6

Simplify and combine like terms.

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Step 6.2.6.1.6.1

Simplify each term.

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Step 6.2.6.1.6.1.1

Multiply $28$ by $28$ .

$y = \frac{- 28 + z \pm \sqrt{784 + 28 (- z) - z \cdot 28 - z (- z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.6.1.2

Multiply $- 1$ by $28$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - z \cdot 28 - z (- z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.6.1.3

Multiply $28$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z - z (- z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.6.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z - 1 \cdot (- 1 z \cdot z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.6.1.5

Multiply $z$ by $z$ by adding the exponents.

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Step 6.2.6.1.6.1.5.1

Move $z$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z - 1 \cdot (- 1 (z \cdot z)) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.6.1.5.2

Multiply $z$ by $z$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z - 1 \cdot (- 1 z^{2}) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.6.1.6

Multiply $- 1$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z + 1 z^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.6.1.7

Multiply $z^{2}$ by $1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z + z^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.6.2

Subtract $28 z$ from $- 28 z$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.7

Multiply $- 4 \cdot - 1 \cdot - 1$ .

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Step 6.2.6.1.7.1

Multiply $- 4$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} + 4 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.6.1.7.2

Multiply $4$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2 \cdot - 1}$

Step 6.2.6.2

Multiply $2$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{- 2}$

Step 6.2.6.3

Simplify $\frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{- 2}$ .

$y = \frac{28 - z \pm \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$

Step 6.2.6.4

Change the $\pm$ to $+$ .

$y = \frac{28 - z + \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$

Step 6.2.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.2.7.1

Simplify the numerator.

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Step 6.2.7.1.1

Apply the distributive property.

$y = \frac{- 1 \cdot 28 + z \pm \sqrt{{(28 - z)}^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.2

Multiply $- 1$ by $28$ .

$y = \frac{- 28 + z \pm \sqrt{{(28 - z)}^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.3

Multiply $- - z$ .

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Step 6.2.7.1.3.1

Multiply $- 1$ by $- 1$ .

$y = \frac{- 28 + 1 z \pm \sqrt{{(28 - z)}^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.3.2

Multiply $z$ by $1$ .

$y = \frac{- 28 + z \pm \sqrt{{(28 - z)}^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.4

Rewrite ${(28 - z)}^{2}$ as $(28 - z) (28 - z)$ .

$y = \frac{- 28 + z \pm \sqrt{(28 - z) (28 - z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.5

Expand $(28 - z) (28 - z)$ using the FOIL Method.

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Step 6.2.7.1.5.1

Apply the distributive property.

$y = \frac{- 28 + z \pm \sqrt{28 (28 - z) - z (28 - z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.5.2

Apply the distributive property.

$y = \frac{- 28 + z \pm \sqrt{28 \cdot 28 + 28 (- z) - z (28 - z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.5.3

Apply the distributive property.

$y = \frac{- 28 + z \pm \sqrt{28 \cdot 28 + 28 (- z) - z \cdot 28 - z (- z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.6

Simplify and combine like terms.

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Step 6.2.7.1.6.1

Simplify each term.

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Step 6.2.7.1.6.1.1

Multiply $28$ by $28$ .

$y = \frac{- 28 + z \pm \sqrt{784 + 28 (- z) - z \cdot 28 - z (- z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.6.1.2

Multiply $- 1$ by $28$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - z \cdot 28 - z (- z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.6.1.3

Multiply $28$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z - z (- z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.6.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z - 1 \cdot (- 1 z \cdot z) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.6.1.5

Multiply $z$ by $z$ by adding the exponents.

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Step 6.2.7.1.6.1.5.1

Move $z$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z - 1 \cdot (- 1 (z \cdot z)) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.6.1.5.2

Multiply $z$ by $z$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z - 1 \cdot (- 1 z^{2}) - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.6.1.6

Multiply $- 1$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z + 1 z^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.6.1.7

Multiply $z^{2}$ by $1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 28 z - 28 z + z^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.6.2

Subtract $28 z$ from $- 28 z$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} - 4 \cdot - 1 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.7

Multiply $- 4 \cdot - 1 \cdot - 1$ .

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Step 6.2.7.1.7.1

Multiply $- 4$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} + 4 \cdot (- 1 z^{2})}}{2 \cdot - 1}$

Step 6.2.7.1.7.2

Multiply $4$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2 \cdot - 1}$

Step 6.2.7.2

Multiply $2$ by $- 1$ .

$y = \frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{- 2}$

Step 6.2.7.3

Simplify $\frac{- 28 + z \pm \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{- 2}$ .

$y = \frac{28 - z \pm \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$

Step 6.2.7.4

Change the $\pm$ to $-$ .

$y = \frac{28 - z - \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$

Step 6.2.8

The final answer is the combination of both solutions.

$y = \frac{28 - z + \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$

$y = \frac{28 - z - \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$

$y = \frac{28 - z + \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$

$y = \frac{28 - z - \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$

$y = \frac{28 - z + \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$

$y = \frac{28 - z - \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$

Step 7

This is the final solution to the system of equations.

$y^{2} = 2128 - 28 y - x^{2} + y^{2} + y z$

$x = 28 - y - z$

$y = \frac{28 - z + \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$

$y = \frac{28 - z - \sqrt{784 - 56 z + z^{2} - 4 z^{2}}}{2}$