Finite Math Examples

$7 x - 8 y = 24$ , $x y^{2} = 1$

Step 1

Divide each term in $x y^{2} = 1$ by $y^{2}$ and simplify.

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Step 1.1

Divide each term in $x y^{2} = 1$ by $y^{2}$ .

$\frac{x y^{2}}{y^{2}} = \frac{1}{y^{2}}$

$7 x - 8 y = 24$

Step 1.2

Simplify the left side.

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Step 1.2.1

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

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Step 1.2.1.1

Cancel the common factor.

$\frac{x y^{2}}{y^{2}} = \frac{1}{y^{2}}$

$7 x - 8 y = 24$

Step 1.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{y^{2}}$

$7 x - 8 y = 24$

$x = \frac{1}{y^{2}}$

$7 x - 8 y = 24$

$x = \frac{1}{y^{2}}$

$7 x - 8 y = 24$

$x = \frac{1}{y^{2}}$

$7 x - 8 y = 24$

Step 2

Replace all occurrences of $x$ with $\frac{1}{y^{2}}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $7 x - 8 y = 24$ with $\frac{1}{y^{2}}$ .

$7 (\frac{1}{y^{2}}) - 8 y = 24$

$x = \frac{1}{y^{2}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Combine $7$ and $\frac{1}{y^{2}}$ .

$\frac{7}{y^{2}} - 8 y = 24$

$x = \frac{1}{y^{2}}$

$\frac{7}{y^{2}} - 8 y = 24$

$x = \frac{1}{y^{2}}$

$\frac{7}{y^{2}} - 8 y = 24$

$x = \frac{1}{y^{2}}$

Step 3

Solve for $y$ in $\frac{7}{y^{2}} - 8 y = 24$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{2}, 1, 1$

$x = \frac{1}{y^{2}}$

Step 3.1.2

The LCM of one and any expression is the expression.

$y^{2}$

$x = \frac{1}{y^{2}}$

$y^{2}$

$x = \frac{1}{y^{2}}$

Step 3.2

Multiply each term in $\frac{7}{y^{2}} - 8 y = 24$ by $y^{2}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{7}{y^{2}} - 8 y = 24$ by $y^{2}$ .

$\frac{7}{y^{2}} \cdot y^{2} - 8 y \cdot y^{2} = 24 y^{2}$

$x = \frac{1}{y^{2}}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

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

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Step 3.2.2.1.1.1

Cancel the common factor.

$\frac{7}{y^{2}} \cdot y^{2} - 8 y \cdot y^{2} = 24 y^{2}$

$x = \frac{1}{y^{2}}$

Step 3.2.2.1.1.2

Rewrite the expression.

$7 - 8 y \cdot y^{2} = 24 y^{2}$

$x = \frac{1}{y^{2}}$

$7 - 8 y \cdot y^{2} = 24 y^{2}$

$x = \frac{1}{y^{2}}$

Step 3.2.2.1.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 3.2.2.1.2.1

Move $y^{2}$ .

$7 - 8 (y^{2} y) = 24 y^{2}$

$x = \frac{1}{y^{2}}$

Step 3.2.2.1.2.2

Multiply $y^{2}$ by $y$ .

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Step 3.2.2.1.2.2.1

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

$7 - 8 (y^{2} y) = 24 y^{2}$

$x = \frac{1}{y^{2}}$

Step 3.2.2.1.2.2.2

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

$7 - 8 y^{2 + 1} = 24 y^{2}$

$x = \frac{1}{y^{2}}$

$7 - 8 y^{2 + 1} = 24 y^{2}$

$x = \frac{1}{y^{2}}$

Step 3.2.2.1.2.3

Add $2$ and $1$ .

$7 - 8 y^{3} = 24 y^{2}$

$x = \frac{1}{y^{2}}$

$7 - 8 y^{3} = 24 y^{2}$

$x = \frac{1}{y^{2}}$

$7 - 8 y^{3} = 24 y^{2}$

$x = \frac{1}{y^{2}}$

$7 - 8 y^{3} = 24 y^{2}$

$x = \frac{1}{y^{2}}$

$7 - 8 y^{3} = 24 y^{2}$

$x = \frac{1}{y^{2}}$

Step 3.3

Solve the equation.

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Step 3.3.1

Subtract $24 y^{2}$ from both sides of the equation.

$7 - 8 y^{3} - 24 y^{2} = 0$

$x = \frac{1}{y^{2}}$

Step 3.3.2

Factor the left side of the equation.

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Step 3.3.2.1

Factor $- 1$ out of $7 - 8 y^{3} - 24 y^{2}$ .

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Step 3.3.2.1.1

Move $7$ .

$- 8 y^{3} - 24 y^{2} + 7 = 0$

$x = \frac{1}{y^{2}}$

Step 3.3.2.1.2

Factor $- 1$ out of $- 8 y^{3}$ .

$- (8 y^{3}) - 24 y^{2} + 7 = 0$

$x = \frac{1}{y^{2}}$

Step 3.3.2.1.3

Factor $- 1$ out of $- 24 y^{2}$ .

$- (8 y^{3}) - (24 y^{2}) + 7 = 0$

$x = \frac{1}{y^{2}}$

Step 3.3.2.1.4

Rewrite $7$ as $- 1 (- 7)$ .

$- (8 y^{3}) - (24 y^{2}) - 1 \cdot - 7 = 0$

$x = \frac{1}{y^{2}}$

Step 3.3.2.1.5

Factor $- 1$ out of $- (8 y^{3}) - (24 y^{2})$ .

$- (8 y^{3} + 24 y^{2}) - 1 \cdot - 7 = 0$

$x = \frac{1}{y^{2}}$

Step 3.3.2.1.6

Factor $- 1$ out of $- (8 y^{3} + 24 y^{2}) - 1 (- 7)$ .

$- (8 y^{3} + 24 y^{2} - 7) = 0$

$x = \frac{1}{y^{2}}$

$- (8 y^{3} + 24 y^{2} - 7) = 0$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2

Factor.

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Step 3.3.2.2.1

Factor $8 y^{3} + 24 y^{2} - 7$ using the rational roots test.

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Step 3.3.2.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 7$

$q = \pm 1, \pm 8, \pm 2, \pm 4$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.125, \pm 0.5, \pm 0.25, \pm 7, \pm 0.875, \pm 3.5, \pm 1.75$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2.1.3

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

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Step 3.3.2.2.1.3.1

Substitute $0.5$ into the polynomial.

$8 \cdot {0.5}^{3} + 24 \cdot {0.5}^{2} - 7$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2.1.3.2

Raise $0.5$ to the power of $3$ .

$8 \cdot 0.125 + 24 \cdot {0.5}^{2} - 7$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2.1.3.3

Multiply $8$ by $0.125$ .

$1 + 24 \cdot {0.5}^{2} - 7$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2.1.3.4

Raise $0.5$ to the power of $2$ .

$1 + 24 \cdot 0.25 - 7$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2.1.3.5

Multiply $24$ by $0.25$ .

$1 + 6 - 7$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2.1.3.6

Add $1$ and $6$ .

$7 - 7$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2.1.3.7

Subtract $7$ from $7$ .

$0$

$x = \frac{1}{y^{2}}$

$0$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2.1.4

Since $0.5$ is a known root, divide the polynomial by $2 y - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{8 y^{3} + 24 y^{2} - 7}{2 y - 1}$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2.1.5

Divide $8 y^{3} + 24 y^{2} - 7$ by $2 y - 1$ .

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Step 3.3.2.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$2 y$

$1$

$8 y^{3}$

$24 y^{2}$

$0 y$

$7$

Step 3.3.2.2.1.5.2

Divide the highest order term in the dividend $8 y^{3}$ by the highest order term in divisor $2 y$ .

					$4 y^{2}$
	$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$

Step 3.3.2.2.1.5.3

Multiply the new quotient term by the divisor.

				$4 y^{2}$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			+	$8 y^{3}$	-	$4 y^{2}$

Step 3.3.2.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $8 y^{3} - 4 y^{2}$

				$4 y^{2}$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			-	$8 y^{3}$	+	$4 y^{2}$

Step 3.3.2.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$4 y^{2}$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			-	$8 y^{3}$	+	$4 y^{2}$
					+	$28 y^{2}$

Step 3.3.2.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$4 y^{2}$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			-	$8 y^{3}$	+	$4 y^{2}$
					+	$28 y^{2}$	+	$0 y$

Step 3.3.2.2.1.5.7

Divide the highest order term in the dividend $28 y^{2}$ by the highest order term in divisor $2 y$ .

				$4 y^{2}$	+	$14 y$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			-	$8 y^{3}$	+	$4 y^{2}$
					+	$28 y^{2}$	+	$0 y$

Step 3.3.2.2.1.5.8

Multiply the new quotient term by the divisor.

				$4 y^{2}$	+	$14 y$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			-	$8 y^{3}$	+	$4 y^{2}$
					+	$28 y^{2}$	+	$0 y$
					+	$28 y^{2}$	-	$14 y$

Step 3.3.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $28 y^{2} - 14 y$

				$4 y^{2}$	+	$14 y$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			-	$8 y^{3}$	+	$4 y^{2}$
					+	$28 y^{2}$	+	$0 y$
					-	$28 y^{2}$	+	$14 y$

Step 3.3.2.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$4 y^{2}$	+	$14 y$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			-	$8 y^{3}$	+	$4 y^{2}$
					+	$28 y^{2}$	+	$0 y$
					-	$28 y^{2}$	+	$14 y$
							+	$14 y$

Step 3.3.2.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$4 y^{2}$	+	$14 y$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			-	$8 y^{3}$	+	$4 y^{2}$
					+	$28 y^{2}$	+	$0 y$
					-	$28 y^{2}$	+	$14 y$
							+	$14 y$	-	$7$

Step 3.3.2.2.1.5.12

Divide the highest order term in the dividend $14 y$ by the highest order term in divisor $2 y$ .

				$4 y^{2}$	+	$14 y$	+	$7$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			-	$8 y^{3}$	+	$4 y^{2}$
					+	$28 y^{2}$	+	$0 y$
					-	$28 y^{2}$	+	$14 y$
							+	$14 y$	-	$7$

Step 3.3.2.2.1.5.13

Multiply the new quotient term by the divisor.

				$4 y^{2}$	+	$14 y$	+	$7$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			-	$8 y^{3}$	+	$4 y^{2}$
					+	$28 y^{2}$	+	$0 y$
					-	$28 y^{2}$	+	$14 y$
							+	$14 y$	-	$7$
							+	$14 y$	-	$7$

Step 3.3.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $14 y - 7$

				$4 y^{2}$	+	$14 y$	+	$7$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			-	$8 y^{3}$	+	$4 y^{2}$
					+	$28 y^{2}$	+	$0 y$
					-	$28 y^{2}$	+	$14 y$
							+	$14 y$	-	$7$
							-	$14 y$	+	$7$

Step 3.3.2.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$4 y^{2}$	+	$14 y$	+	$7$
$2 y$	-	$1$		$8 y^{3}$	+	$24 y^{2}$	+	$0 y$	-	$7$
			-	$8 y^{3}$	+	$4 y^{2}$
					+	$28 y^{2}$	+	$0 y$
					-	$28 y^{2}$	+	$14 y$
							+	$14 y$	-	$7$
							-	$14 y$	+	$7$
										$0$

Step 3.3.2.2.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$4 y^{2} + 14 y + 7$

$x = \frac{1}{y^{2}}$

$4 y^{2} + 14 y + 7$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2.1.6

Write $8 y^{3} + 24 y^{2} - 7$ as a set of factors.

$- ((2 y - 1) (4 y^{2} + 14 y + 7)) = 0$

$x = \frac{1}{y^{2}}$

$- ((2 y - 1) (4 y^{2} + 14 y + 7)) = 0$

$x = \frac{1}{y^{2}}$

Step 3.3.2.2.2

Remove unnecessary parentheses.

$- (2 y - 1) (4 y^{2} + 14 y + 7) = 0$

$x = \frac{1}{y^{2}}$

$- (2 y - 1) (4 y^{2} + 14 y + 7) = 0$

$x = \frac{1}{y^{2}}$

$- (2 y - 1) (4 y^{2} + 14 y + 7) = 0$

$x = \frac{1}{y^{2}}$

Step 3.3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 y - 1 = 0$

$4 y^{2} + 14 y + 7 = 0$

$x = \frac{1}{y^{2}}$

Step 3.3.4

Set $2 y - 1$ equal to $0$ and solve for $y$ .

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Step 3.3.4.1

Set $2 y - 1$ equal to $0$ .

$2 y - 1 = 0$

$x = \frac{1}{y^{2}}$

Step 3.3.4.2

Solve $2 y - 1 = 0$ for $y$ .

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Step 3.3.4.2.1

Add $1$ to both sides of the equation.

$2 y = 1$

$x = \frac{1}{y^{2}}$

Step 3.3.4.2.2

Divide each term in $2 y = 1$ by $2$ and simplify.

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Step 3.3.4.2.2.1

Divide each term in $2 y = 1$ by $2$ .

$\frac{2 y}{2} = \frac{1}{2}$

$x = \frac{1}{y^{2}}$

Step 3.3.4.2.2.2

Simplify the left side.

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Step 3.3.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.3.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{1}{2}$

$x = \frac{1}{y^{2}}$

Step 3.3.4.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{2}$

$x = \frac{1}{y^{2}}$

$y = \frac{1}{2}$

$x = \frac{1}{y^{2}}$

$y = \frac{1}{2}$

$x = \frac{1}{y^{2}}$

$y = \frac{1}{2}$

$x = \frac{1}{y^{2}}$

$y = \frac{1}{2}$

$x = \frac{1}{y^{2}}$

$y = \frac{1}{2}$

$x = \frac{1}{y^{2}}$

Step 3.3.5

Set $4 y^{2} + 14 y + 7$ equal to $0$ and solve for $y$ .

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Step 3.3.5.1

Set $4 y^{2} + 14 y + 7$ equal to $0$ .

$4 y^{2} + 14 y + 7 = 0$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2

Solve $4 y^{2} + 14 y + 7 = 0$ for $y$ .

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Step 3.3.5.2.1

Use the quadratic formula to find the solutions.

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

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.2

Substitute the values $a = 4$ , $b = 14$ , and $c = 7$ into the quadratic formula and solve for $y$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (4 \cdot 7)}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.3

Simplify.

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Step 3.3.5.2.3.1

Simplify the numerator.

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Step 3.3.5.2.3.1.1

Raise $14$ to the power of $2$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot 4 \cdot 7}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.3.1.2

Multiply $- 4 \cdot 4 \cdot 7$ .

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Step 3.3.5.2.3.1.2.1

Multiply $- 4$ by $4$ .

$y = \frac{- 14 \pm \sqrt{196 - 16 \cdot 7}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.3.1.2.2

Multiply $- 16$ by $7$ .

$y = \frac{- 14 \pm \sqrt{196 - 112}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

$y = \frac{- 14 \pm \sqrt{196 - 112}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.3.1.3

Subtract $112$ from $196$ .

$y = \frac{- 14 \pm \sqrt{84}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.3.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 3.3.5.2.3.1.4.1

Factor $4$ out of $84$ .

$y = \frac{- 14 \pm \sqrt{4 (21)}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.3.1.4.2

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

$y = \frac{- 14 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

$y = \frac{- 14 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.3.1.5

Pull terms out from under the radical.

$y = \frac{- 14 \pm 2 \sqrt{21}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

$y = \frac{- 14 \pm 2 \sqrt{21}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.3.2

Multiply $2$ by $4$ .

$y = \frac{- 14 \pm 2 \sqrt{21}}{8}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.3.3

Simplify $\frac{- 14 \pm 2 \sqrt{21}}{8}$ .

$y = \frac{- 7 \pm \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

$y = \frac{- 7 \pm \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4

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

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Step 3.3.5.2.4.1

Simplify the numerator.

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Step 3.3.5.2.4.1.1

Raise $14$ to the power of $2$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot 4 \cdot 7}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.1.2

Multiply $- 4 \cdot 4 \cdot 7$ .

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Step 3.3.5.2.4.1.2.1

Multiply $- 4$ by $4$ .

$y = \frac{- 14 \pm \sqrt{196 - 16 \cdot 7}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.1.2.2

Multiply $- 16$ by $7$ .

$y = \frac{- 14 \pm \sqrt{196 - 112}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

$y = \frac{- 14 \pm \sqrt{196 - 112}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.1.3

Subtract $112$ from $196$ .

$y = \frac{- 14 \pm \sqrt{84}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 3.3.5.2.4.1.4.1

Factor $4$ out of $84$ .

$y = \frac{- 14 \pm \sqrt{4 (21)}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.1.4.2

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

$y = \frac{- 14 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

$y = \frac{- 14 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.1.5

Pull terms out from under the radical.

$y = \frac{- 14 \pm 2 \sqrt{21}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

$y = \frac{- 14 \pm 2 \sqrt{21}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.2

Multiply $2$ by $4$ .

$y = \frac{- 14 \pm 2 \sqrt{21}}{8}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.3

Simplify $\frac{- 14 \pm 2 \sqrt{21}}{8}$ .

$y = \frac{- 7 \pm \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.4

Change the $\pm$ to $+$ .

$y = \frac{- 7 + \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.5

Rewrite $- 7$ as $- 1 (7)$ .

$y = \frac{- 1 \cdot 7 + \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.6

Factor $- 1$ out of $\sqrt{21}$ .

$y = \frac{- 1 \cdot 7 - 1 (- \sqrt{21})}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.7

Factor $- 1$ out of $- 1 (7) - 1 (- \sqrt{21})$ .

$y = \frac{- 1 (7 - \sqrt{21})}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.4.8

Move the negative in front of the fraction.

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5

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

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Step 3.3.5.2.5.1

Simplify the numerator.

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Step 3.3.5.2.5.1.1

Raise $14$ to the power of $2$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot 4 \cdot 7}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.1.2

Multiply $- 4 \cdot 4 \cdot 7$ .

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Step 3.3.5.2.5.1.2.1

Multiply $- 4$ by $4$ .

$y = \frac{- 14 \pm \sqrt{196 - 16 \cdot 7}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.1.2.2

Multiply $- 16$ by $7$ .

$y = \frac{- 14 \pm \sqrt{196 - 112}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

$y = \frac{- 14 \pm \sqrt{196 - 112}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.1.3

Subtract $112$ from $196$ .

$y = \frac{- 14 \pm \sqrt{84}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 3.3.5.2.5.1.4.1

Factor $4$ out of $84$ .

$y = \frac{- 14 \pm \sqrt{4 (21)}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.1.4.2

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

$y = \frac{- 14 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

$y = \frac{- 14 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.1.5

Pull terms out from under the radical.

$y = \frac{- 14 \pm 2 \sqrt{21}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

$y = \frac{- 14 \pm 2 \sqrt{21}}{2 \cdot 4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.2

Multiply $2$ by $4$ .

$y = \frac{- 14 \pm 2 \sqrt{21}}{8}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.3

Simplify $\frac{- 14 \pm 2 \sqrt{21}}{8}$ .

$y = \frac{- 7 \pm \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.4

Change the $\pm$ to $-$ .

$y = \frac{- 7 - \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.5

Rewrite $- 7$ as $- 1 (7)$ .

$y = \frac{- 1 \cdot 7 - \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.6

Factor $- 1$ out of $- \sqrt{21}$ .

$y = \frac{- 1 \cdot 7 - (\sqrt{21})}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.7

Factor $- 1$ out of $- 1 (7) - (\sqrt{21})$ .

$y = \frac{- 1 (7 + \sqrt{21})}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.5.8

Move the negative in front of the fraction.

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.5.2.6

The final answer is the combination of both solutions.

$y = - \frac{7 - \sqrt{21}}{4}, - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

$y = - \frac{7 - \sqrt{21}}{4}, - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

$y = - \frac{7 - \sqrt{21}}{4}, - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

Step 3.3.6

The final solution is all the values that make $- (2 y - 1) (4 y^{2} + 14 y + 7) = 0$ true.

$y = \frac{1}{2}, - \frac{7 - \sqrt{21}}{4}, - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

$y = \frac{1}{2}, - \frac{7 - \sqrt{21}}{4}, - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

$y = \frac{1}{2}, - \frac{7 - \sqrt{21}}{4}, - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{y^{2}}$

Step 4

Replace all occurrences of $y$ with $\frac{1}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{1}{y^{2}}$ with $\frac{1}{2}$ .

$x = \frac{1}{{(\frac{1}{2})}^{2}}$

$y = \frac{1}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{1}{{(\frac{1}{2})}^{2}}$ .

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Step 4.2.1.1

Simplify the denominator.

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Step 4.2.1.1.1

Apply the product rule to $\frac{1}{2}$ .

$x = \frac{1}{\frac{1^{2}}{2^{2}}}$

$y = \frac{1}{2}$

Step 4.2.1.1.2

One to any power is one.

$x = \frac{1}{\frac{1}{2^{2}}}$

$y = \frac{1}{2}$

Step 4.2.1.1.3

Raise $2$ to the power of $2$ .

$x = \frac{1}{\frac{1}{4}}$

$y = \frac{1}{2}$

$x = \frac{1}{\frac{1}{4}}$

$y = \frac{1}{2}$

Step 4.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = 1 \cdot 4$

$y = \frac{1}{2}$

Step 4.2.1.3

Multiply $4$ by $1$ .

$x = 4$

$y = \frac{1}{2}$

$x = 4$

$y = \frac{1}{2}$

$x = 4$

$y = \frac{1}{2}$

$x = 4$

$y = \frac{1}{2}$

Step 5

Replace all occurrences of $y$ with $- \frac{7 - \sqrt{21}}{4}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{1}{y^{2}}$ with $- \frac{7 - \sqrt{21}}{4}$ .

$x = \frac{1}{{(- \frac{7 - \sqrt{21}}{4})}^{2}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{1}{{(- \frac{7 - \sqrt{21}}{4})}^{2}}$ .

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Step 5.2.1.1

Simplify the denominator.

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Step 5.2.1.1.1

Apply the product rule to $- \frac{7 - \sqrt{21}}{4}$ .

$x = \frac{1}{{(- 1)}^{2} {(\frac{7 - \sqrt{21}}{4})}^{2}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.2

Raise $- 1$ to the power of $2$ .

$x = \frac{1}{1 {(\frac{7 - \sqrt{21}}{4})}^{2}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.3

Apply the product rule to $\frac{7 - \sqrt{21}}{4}$ .

$x = \frac{1}{1 (\frac{{(7 - \sqrt{21})}^{2}}{4^{2}})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.4

Raise $4$ to the power of $2$ .

$x = \frac{1}{1 (\frac{{(7 - \sqrt{21})}^{2}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.5

Rewrite ${(7 - \sqrt{21})}^{2}$ as $(7 - \sqrt{21}) (7 - \sqrt{21})$ .

$x = \frac{1}{1 (\frac{(7 - \sqrt{21}) (7 - \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.6

Expand $(7 - \sqrt{21}) (7 - \sqrt{21})$ using the FOIL Method.

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Step 5.2.1.1.6.1

Apply the distributive property.

$x = \frac{1}{1 (\frac{7 (7 - \sqrt{21}) - \sqrt{21} (7 - \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.6.2

Apply the distributive property.

$x = \frac{1}{1 (\frac{7 \cdot 7 + 7 (- \sqrt{21}) - \sqrt{21} (7 - \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.6.3

Apply the distributive property.

$x = \frac{1}{1 (\frac{7 \cdot 7 + 7 (- \sqrt{21}) - \sqrt{21} \cdot 7 - \sqrt{21} (- \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{7 \cdot 7 + 7 (- \sqrt{21}) - \sqrt{21} \cdot 7 - \sqrt{21} (- \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7

Simplify and combine like terms.

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Step 5.2.1.1.7.1

Simplify each term.

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Step 5.2.1.1.7.1.1

Multiply $7$ by $7$ .

$x = \frac{1}{1 (\frac{49 + 7 (- \sqrt{21}) - \sqrt{21} \cdot 7 - \sqrt{21} (- \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.2

Multiply $- 1$ by $7$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - \sqrt{21} \cdot 7 - \sqrt{21} (- \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.3

Multiply $7$ by $- 1$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} - \sqrt{21} (- \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.4

Multiply $- \sqrt{21} (- \sqrt{21})$ .

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Step 5.2.1.1.7.1.4.1

Multiply $- 1$ by $- 1$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 1 \sqrt{21} \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.4.2

Multiply $\sqrt{21}$ by $1$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + \sqrt{21} \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.4.3

Raise $\sqrt{21}$ to the power of $1$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + \sqrt{21} \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.4.4

Raise $\sqrt{21}$ to the power of $1$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + \sqrt{21} \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.4.5

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

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + {\sqrt{21}}^{1 + 1}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.4.6

Add $1$ and $1$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + {\sqrt{21}}^{2}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + {\sqrt{21}}^{2}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.5

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 5.2.1.1.7.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + {(21^{\frac{1}{2}})}^{2}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.5.2

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

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21^{\frac{1}{2} \cdot 2}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.5.3

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

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21^{\frac{2}{2}}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.5.4

Cancel the common factor of $2$ .

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Step 5.2.1.1.7.1.5.4.1

Cancel the common factor.

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21^{\frac{2}{2}}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.5.4.2

Rewrite the expression.

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.1.5.5

Evaluate the exponent.

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.2

Add $49$ and $21$ .

$x = \frac{1}{1 (\frac{70 - 7 \sqrt{21} - 7 \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.7.3

Subtract $7 \sqrt{21}$ from $- 7 \sqrt{21}$ .

$x = \frac{1}{1 (\frac{70 - 14 \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{70 - 14 \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.8

Cancel the common factor of $70 - 14 \sqrt{21}$ and $16$ .

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Step 5.2.1.1.8.1

Factor $2$ out of $70$ .

$x = \frac{1}{1 (\frac{2 (35) - 14 \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.8.2

Factor $2$ out of $- 14 \sqrt{21}$ .

$x = \frac{1}{1 (\frac{2 (35) + 2 (- 7 \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.8.3

Factor $2$ out of $2 (35) + 2 (- 7 \sqrt{21})$ .

$x = \frac{1}{1 (\frac{2 (35 - 7 \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.8.4

Cancel the common factors.

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Step 5.2.1.1.8.4.1

Factor $2$ out of $16$ .

$x = \frac{1}{1 (\frac{2 (35 - 7 \sqrt{21})}{2 \cdot 8})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.8.4.2

Cancel the common factor.

$x = \frac{1}{1 (\frac{2 (35 - 7 \sqrt{21})}{2 \cdot 8})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.8.4.3

Rewrite the expression.

$x = \frac{1}{1 (\frac{35 - 7 \sqrt{21}}{8})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{35 - 7 \sqrt{21}}{8})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{35 - 7 \sqrt{21}}{8})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.1.9

Multiply $\frac{35 - 7 \sqrt{21}}{8}$ by $1$ .

$x = \frac{1}{\frac{35 - 7 \sqrt{21}}{8}}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{\frac{35 - 7 \sqrt{21}}{8}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = 1 (\frac{8}{35 - 7 \sqrt{21}})$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.3

Multiply $\frac{8}{35 - 7 \sqrt{21}}$ by $1$ .

$x = \frac{8}{35 - 7 \sqrt{21}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.4

Multiply $\frac{8}{35 - 7 \sqrt{21}}$ by $\frac{35 + 7 \sqrt{21}}{35 + 7 \sqrt{21}}$ .

$x = \frac{8}{35 - 7 \sqrt{21}} \cdot \frac{35 + 7 \sqrt{21}}{35 + 7 \sqrt{21}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.5

Multiply $\frac{8}{35 - 7 \sqrt{21}}$ by $\frac{35 + 7 \sqrt{21}}{35 + 7 \sqrt{21}}$ .

$x = \frac{8 (35 + 7 \sqrt{21})}{(35 - 7 \sqrt{21}) (35 + 7 \sqrt{21})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.6

Expand the denominator using the FOIL method.

$x = \frac{8 (35 + 7 \sqrt{21})}{1225 + 245 \sqrt{21} - 245 \sqrt{21} - 49 {\sqrt{21}}^{2}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.7

Simplify.

$x = \frac{8 (35 + 7 \sqrt{21})}{196}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.8

Cancel the common factor of $8$ and $196$ .

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Step 5.2.1.8.1

Factor $4$ out of $8 (35 + 7 \sqrt{21})$ .

$x = \frac{4 (2 (35 + 7 \sqrt{21}))}{196}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.8.2

Cancel the common factors.

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Step 5.2.1.8.2.1

Factor $4$ out of $196$ .

$x = \frac{4 (2 (35 + 7 \sqrt{21}))}{4 (49)}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.8.2.2

Cancel the common factor.

$x = \frac{4 (2 (35 + 7 \sqrt{21}))}{4 \cdot 49}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.8.2.3

Rewrite the expression.

$x = \frac{2 (35 + 7 \sqrt{21})}{49}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (35 + 7 \sqrt{21})}{49}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (35 + 7 \sqrt{21})}{49}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.9

Cancel the common factor of $35 + 7 \sqrt{21}$ and $49$ .

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Step 5.2.1.9.1

Factor $7$ out of $2 (35 + 7 \sqrt{21})$ .

$x = \frac{7 (2 (5 + \sqrt{21}))}{49}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.9.2

Cancel the common factors.

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Step 5.2.1.9.2.1

Factor $7$ out of $49$ .

$x = \frac{7 (2 (5 + \sqrt{21}))}{7 (7)}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.9.2.2

Cancel the common factor.

$x = \frac{7 (2 (5 + \sqrt{21}))}{7 \cdot 7}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 5.2.1.9.2.3

Rewrite the expression.

$x = \frac{2 (5 + \sqrt{21})}{7}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (5 + \sqrt{21})}{7}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (5 + \sqrt{21})}{7}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (5 + \sqrt{21})}{7}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (5 + \sqrt{21})}{7}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (5 + \sqrt{21})}{7}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 6

Replace all occurrences of $y$ with $\frac{1}{2}$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = \frac{1}{y^{2}}$ with $\frac{1}{2}$ .

$x = \frac{1}{{(\frac{1}{2})}^{2}}$

$y = \frac{1}{2}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{1}{{(\frac{1}{2})}^{2}}$ .

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Step 6.2.1.1

Simplify the denominator.

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Step 6.2.1.1.1

Apply the product rule to $\frac{1}{2}$ .

$x = \frac{1}{\frac{1^{2}}{2^{2}}}$

$y = \frac{1}{2}$

Step 6.2.1.1.2

One to any power is one.

$x = \frac{1}{\frac{1}{2^{2}}}$

$y = \frac{1}{2}$

Step 6.2.1.1.3

Raise $2$ to the power of $2$ .

$x = \frac{1}{\frac{1}{4}}$

$y = \frac{1}{2}$

$x = \frac{1}{\frac{1}{4}}$

$y = \frac{1}{2}$

Step 6.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = 1 \cdot 4$

$y = \frac{1}{2}$

Step 6.2.1.3

Multiply $4$ by $1$ .

$x = 4$

$y = \frac{1}{2}$

$x = 4$

$y = \frac{1}{2}$

$x = 4$

$y = \frac{1}{2}$

$x = 4$

$y = \frac{1}{2}$

Step 7

Replace all occurrences of $y$ with $- \frac{7 - \sqrt{21}}{4}$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $x = \frac{1}{y^{2}}$ with $- \frac{7 - \sqrt{21}}{4}$ .

$x = \frac{1}{{(- \frac{7 - \sqrt{21}}{4})}^{2}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{1}{{(- \frac{7 - \sqrt{21}}{4})}^{2}}$ .

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Step 7.2.1.1

Simplify the denominator.

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Step 7.2.1.1.1

Apply the product rule to $- \frac{7 - \sqrt{21}}{4}$ .

$x = \frac{1}{{(- 1)}^{2} {(\frac{7 - \sqrt{21}}{4})}^{2}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.2

Raise $- 1$ to the power of $2$ .

$x = \frac{1}{1 {(\frac{7 - \sqrt{21}}{4})}^{2}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.3

Apply the product rule to $\frac{7 - \sqrt{21}}{4}$ .

$x = \frac{1}{1 (\frac{{(7 - \sqrt{21})}^{2}}{4^{2}})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.4

Raise $4$ to the power of $2$ .

$x = \frac{1}{1 (\frac{{(7 - \sqrt{21})}^{2}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.5

Rewrite ${(7 - \sqrt{21})}^{2}$ as $(7 - \sqrt{21}) (7 - \sqrt{21})$ .

$x = \frac{1}{1 (\frac{(7 - \sqrt{21}) (7 - \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.6

Expand $(7 - \sqrt{21}) (7 - \sqrt{21})$ using the FOIL Method.

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Step 7.2.1.1.6.1

Apply the distributive property.

$x = \frac{1}{1 (\frac{7 (7 - \sqrt{21}) - \sqrt{21} (7 - \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.6.2

Apply the distributive property.

$x = \frac{1}{1 (\frac{7 \cdot 7 + 7 (- \sqrt{21}) - \sqrt{21} (7 - \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.6.3

Apply the distributive property.

$x = \frac{1}{1 (\frac{7 \cdot 7 + 7 (- \sqrt{21}) - \sqrt{21} \cdot 7 - \sqrt{21} (- \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{7 \cdot 7 + 7 (- \sqrt{21}) - \sqrt{21} \cdot 7 - \sqrt{21} (- \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7

Simplify and combine like terms.

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Step 7.2.1.1.7.1

Simplify each term.

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Step 7.2.1.1.7.1.1

Multiply $7$ by $7$ .

$x = \frac{1}{1 (\frac{49 + 7 (- \sqrt{21}) - \sqrt{21} \cdot 7 - \sqrt{21} (- \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.2

Multiply $- 1$ by $7$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - \sqrt{21} \cdot 7 - \sqrt{21} (- \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.3

Multiply $7$ by $- 1$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} - \sqrt{21} (- \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.4

Multiply $- \sqrt{21} (- \sqrt{21})$ .

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Step 7.2.1.1.7.1.4.1

Multiply $- 1$ by $- 1$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 1 \sqrt{21} \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.4.2

Multiply $\sqrt{21}$ by $1$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + \sqrt{21} \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.4.3

Raise $\sqrt{21}$ to the power of $1$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + \sqrt{21} \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.4.4

Raise $\sqrt{21}$ to the power of $1$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + \sqrt{21} \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.4.5

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

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + {\sqrt{21}}^{1 + 1}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.4.6

Add $1$ and $1$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + {\sqrt{21}}^{2}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + {\sqrt{21}}^{2}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.5

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 7.2.1.1.7.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + {(21^{\frac{1}{2}})}^{2}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.5.2

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

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21^{\frac{1}{2} \cdot 2}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.5.3

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

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21^{\frac{2}{2}}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.5.4

Cancel the common factor of $2$ .

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Step 7.2.1.1.7.1.5.4.1

Cancel the common factor.

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21^{\frac{2}{2}}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.5.4.2

Rewrite the expression.

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.1.5.5

Evaluate the exponent.

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{49 - 7 \sqrt{21} - 7 \sqrt{21} + 21}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.2

Add $49$ and $21$ .

$x = \frac{1}{1 (\frac{70 - 7 \sqrt{21} - 7 \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.7.3

Subtract $7 \sqrt{21}$ from $- 7 \sqrt{21}$ .

$x = \frac{1}{1 (\frac{70 - 14 \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{70 - 14 \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.8

Cancel the common factor of $70 - 14 \sqrt{21}$ and $16$ .

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Step 7.2.1.1.8.1

Factor $2$ out of $70$ .

$x = \frac{1}{1 (\frac{2 (35) - 14 \sqrt{21}}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.8.2

Factor $2$ out of $- 14 \sqrt{21}$ .

$x = \frac{1}{1 (\frac{2 (35) + 2 (- 7 \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.8.3

Factor $2$ out of $2 (35) + 2 (- 7 \sqrt{21})$ .

$x = \frac{1}{1 (\frac{2 (35 - 7 \sqrt{21})}{16})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.8.4

Cancel the common factors.

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Step 7.2.1.1.8.4.1

Factor $2$ out of $16$ .

$x = \frac{1}{1 (\frac{2 (35 - 7 \sqrt{21})}{2 \cdot 8})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.8.4.2

Cancel the common factor.

$x = \frac{1}{1 (\frac{2 (35 - 7 \sqrt{21})}{2 \cdot 8})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.8.4.3

Rewrite the expression.

$x = \frac{1}{1 (\frac{35 - 7 \sqrt{21}}{8})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{35 - 7 \sqrt{21}}{8})}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{35 - 7 \sqrt{21}}{8})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.1.9

Multiply $\frac{35 - 7 \sqrt{21}}{8}$ by $1$ .

$x = \frac{1}{\frac{35 - 7 \sqrt{21}}{8}}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{1}{\frac{35 - 7 \sqrt{21}}{8}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = 1 (\frac{8}{35 - 7 \sqrt{21}})$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.3

Multiply $\frac{8}{35 - 7 \sqrt{21}}$ by $1$ .

$x = \frac{8}{35 - 7 \sqrt{21}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.4

Multiply $\frac{8}{35 - 7 \sqrt{21}}$ by $\frac{35 + 7 \sqrt{21}}{35 + 7 \sqrt{21}}$ .

$x = \frac{8}{35 - 7 \sqrt{21}} \cdot \frac{35 + 7 \sqrt{21}}{35 + 7 \sqrt{21}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.5

Multiply $\frac{8}{35 - 7 \sqrt{21}}$ by $\frac{35 + 7 \sqrt{21}}{35 + 7 \sqrt{21}}$ .

$x = \frac{8 (35 + 7 \sqrt{21})}{(35 - 7 \sqrt{21}) (35 + 7 \sqrt{21})}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.6

Expand the denominator using the FOIL method.

$x = \frac{8 (35 + 7 \sqrt{21})}{1225 + 245 \sqrt{21} - 245 \sqrt{21} - 49 {\sqrt{21}}^{2}}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.7

Simplify.

$x = \frac{8 (35 + 7 \sqrt{21})}{196}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.8

Cancel the common factor of $8$ and $196$ .

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Step 7.2.1.8.1

Factor $4$ out of $8 (35 + 7 \sqrt{21})$ .

$x = \frac{4 (2 (35 + 7 \sqrt{21}))}{196}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.8.2

Cancel the common factors.

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Step 7.2.1.8.2.1

Factor $4$ out of $196$ .

$x = \frac{4 (2 (35 + 7 \sqrt{21}))}{4 (49)}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.8.2.2

Cancel the common factor.

$x = \frac{4 (2 (35 + 7 \sqrt{21}))}{4 \cdot 49}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.8.2.3

Rewrite the expression.

$x = \frac{2 (35 + 7 \sqrt{21})}{49}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (35 + 7 \sqrt{21})}{49}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (35 + 7 \sqrt{21})}{49}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.9

Cancel the common factor of $35 + 7 \sqrt{21}$ and $49$ .

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Step 7.2.1.9.1

Factor $7$ out of $2 (35 + 7 \sqrt{21})$ .

$x = \frac{7 (2 (5 + \sqrt{21}))}{49}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.9.2

Cancel the common factors.

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Step 7.2.1.9.2.1

Factor $7$ out of $49$ .

$x = \frac{7 (2 (5 + \sqrt{21}))}{7 (7)}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.9.2.2

Cancel the common factor.

$x = \frac{7 (2 (5 + \sqrt{21}))}{7 \cdot 7}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 7.2.1.9.2.3

Rewrite the expression.

$x = \frac{2 (5 + \sqrt{21})}{7}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (5 + \sqrt{21})}{7}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (5 + \sqrt{21})}{7}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (5 + \sqrt{21})}{7}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (5 + \sqrt{21})}{7}$

$y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (5 + \sqrt{21})}{7}$

$y = - \frac{7 - \sqrt{21}}{4}$

Step 8

Replace all occurrences of $y$ with $- \frac{7 + \sqrt{21}}{4}$ in each equation.

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Step 8.1

Replace all occurrences of $y$ in $x = \frac{1}{y^{2}}$ with $- \frac{7 + \sqrt{21}}{4}$ .

$x = \frac{1}{{(- \frac{7 + \sqrt{21}}{4})}^{2}}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{1}{{(- \frac{7 + \sqrt{21}}{4})}^{2}}$ .

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Step 8.2.1.1

Simplify the denominator.

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Step 8.2.1.1.1

Apply the product rule to $- \frac{7 + \sqrt{21}}{4}$ .

$x = \frac{1}{{(- 1)}^{2} {(\frac{7 + \sqrt{21}}{4})}^{2}}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.2

Raise $- 1$ to the power of $2$ .

$x = \frac{1}{1 {(\frac{7 + \sqrt{21}}{4})}^{2}}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.3

Apply the product rule to $\frac{7 + \sqrt{21}}{4}$ .

$x = \frac{1}{1 (\frac{{(7 + \sqrt{21})}^{2}}{4^{2}})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.4

Raise $4$ to the power of $2$ .

$x = \frac{1}{1 (\frac{{(7 + \sqrt{21})}^{2}}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.5

Rewrite ${(7 + \sqrt{21})}^{2}$ as $(7 + \sqrt{21}) (7 + \sqrt{21})$ .

$x = \frac{1}{1 (\frac{(7 + \sqrt{21}) (7 + \sqrt{21})}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.6

Expand $(7 + \sqrt{21}) (7 + \sqrt{21})$ using the FOIL Method.

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Step 8.2.1.1.6.1

Apply the distributive property.

$x = \frac{1}{1 (\frac{7 (7 + \sqrt{21}) + \sqrt{21} (7 + \sqrt{21})}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.6.2

Apply the distributive property.

$x = \frac{1}{1 (\frac{7 \cdot 7 + 7 \sqrt{21} + \sqrt{21} (7 + \sqrt{21})}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.6.3

Apply the distributive property.

$x = \frac{1}{1 (\frac{7 \cdot 7 + 7 \sqrt{21} + \sqrt{21} \cdot 7 + \sqrt{21} \sqrt{21}}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{7 \cdot 7 + 7 \sqrt{21} + \sqrt{21} \cdot 7 + \sqrt{21} \sqrt{21}}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.7

Simplify and combine like terms.

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Step 8.2.1.1.7.1

Simplify each term.

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Step 8.2.1.1.7.1.1

Multiply $7$ by $7$ .

$x = \frac{1}{1 (\frac{49 + 7 \sqrt{21} + \sqrt{21} \cdot 7 + \sqrt{21} \sqrt{21}}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.7.1.2

Move $7$ to the left of $\sqrt{21}$ .

$x = \frac{1}{1 (\frac{49 + 7 \sqrt{21} + 7 \cdot \sqrt{21} + \sqrt{21} \sqrt{21}}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.7.1.3

Combine using the product rule for radicals.

$x = \frac{1}{1 (\frac{49 + 7 \sqrt{21} + 7 \sqrt{21} + \sqrt{21 \cdot 21}}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.7.1.4

Multiply $21$ by $21$ .

$x = \frac{1}{1 (\frac{49 + 7 \sqrt{21} + 7 \sqrt{21} + \sqrt{441}}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.7.1.5

Rewrite $441$ as $21^{2}$ .

$x = \frac{1}{1 (\frac{49 + 7 \sqrt{21} + 7 \sqrt{21} + \sqrt{21^{2}}}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.7.1.6

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

$x = \frac{1}{1 (\frac{49 + 7 \sqrt{21} + 7 \sqrt{21} + 21}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{49 + 7 \sqrt{21} + 7 \sqrt{21} + 21}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.7.2

Add $49$ and $21$ .

$x = \frac{1}{1 (\frac{70 + 7 \sqrt{21} + 7 \sqrt{21}}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.7.3

Add $7 \sqrt{21}$ and $7 \sqrt{21}$ .

$x = \frac{1}{1 (\frac{70 + 14 \sqrt{21}}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{70 + 14 \sqrt{21}}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.8

Cancel the common factor of $70 + 14 \sqrt{21}$ and $16$ .

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Step 8.2.1.1.8.1

Factor $2$ out of $70$ .

$x = \frac{1}{1 (\frac{2 (35) + 14 \sqrt{21}}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.8.2

Factor $2$ out of $14 \sqrt{21}$ .

$x = \frac{1}{1 (\frac{2 (35) + 2 (7 \sqrt{21})}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.8.3

Factor $2$ out of $2 (35) + 2 (7 \sqrt{21})$ .

$x = \frac{1}{1 (\frac{2 (35 + 7 \sqrt{21})}{16})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.8.4

Cancel the common factors.

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Step 8.2.1.1.8.4.1

Factor $2$ out of $16$ .

$x = \frac{1}{1 (\frac{2 (35 + 7 \sqrt{21})}{2 \cdot 8})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.8.4.2

Cancel the common factor.

$x = \frac{1}{1 (\frac{2 (35 + 7 \sqrt{21})}{2 \cdot 8})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.8.4.3

Rewrite the expression.

$x = \frac{1}{1 (\frac{35 + 7 \sqrt{21}}{8})}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{35 + 7 \sqrt{21}}{8})}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{1 (\frac{35 + 7 \sqrt{21}}{8})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.1.9

Multiply $\frac{35 + 7 \sqrt{21}}{8}$ by $1$ .

$x = \frac{1}{\frac{35 + 7 \sqrt{21}}{8}}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{1}{\frac{35 + 7 \sqrt{21}}{8}}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = 1 (\frac{8}{35 + 7 \sqrt{21}})$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.3

Multiply $\frac{8}{35 + 7 \sqrt{21}}$ by $1$ .

$x = \frac{8}{35 + 7 \sqrt{21}}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.4

Multiply $\frac{8}{35 + 7 \sqrt{21}}$ by $\frac{35 - 7 \sqrt{21}}{35 - 7 \sqrt{21}}$ .

$x = \frac{8}{35 + 7 \sqrt{21}} \cdot \frac{35 - 7 \sqrt{21}}{35 - 7 \sqrt{21}}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.5

Multiply $\frac{8}{35 + 7 \sqrt{21}}$ by $\frac{35 - 7 \sqrt{21}}{35 - 7 \sqrt{21}}$ .

$x = \frac{8 (35 - 7 \sqrt{21})}{(35 + 7 \sqrt{21}) (35 - 7 \sqrt{21})}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.6

Expand the denominator using the FOIL method.

$x = \frac{8 (35 - 7 \sqrt{21})}{1225 - 245 \sqrt{21} + 245 \sqrt{21} - 49 {\sqrt{21}}^{2}}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.7

Simplify.

$x = \frac{8 (35 - 7 \sqrt{21})}{196}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.8

Cancel the common factor of $8$ and $196$ .

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Step 8.2.1.8.1

Factor $4$ out of $8 (35 - 7 \sqrt{21})$ .

$x = \frac{4 (2 (35 - 7 \sqrt{21}))}{196}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.8.2

Cancel the common factors.

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Step 8.2.1.8.2.1

Factor $4$ out of $196$ .

$x = \frac{4 (2 (35 - 7 \sqrt{21}))}{4 (49)}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.8.2.2

Cancel the common factor.

$x = \frac{4 (2 (35 - 7 \sqrt{21}))}{4 \cdot 49}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.8.2.3

Rewrite the expression.

$x = \frac{2 (35 - 7 \sqrt{21})}{49}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{2 (35 - 7 \sqrt{21})}{49}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{2 (35 - 7 \sqrt{21})}{49}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.9

Cancel the common factor of $35 - 7 \sqrt{21}$ and $49$ .

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Step 8.2.1.9.1

Factor $7$ out of $2 (35 - 7 \sqrt{21})$ .

$x = \frac{7 (2 (5 - \sqrt{21}))}{49}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.9.2

Cancel the common factors.

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Step 8.2.1.9.2.1

Factor $7$ out of $49$ .

$x = \frac{7 (2 (5 - \sqrt{21}))}{7 (7)}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.9.2.2

Cancel the common factor.

$x = \frac{7 (2 (5 - \sqrt{21}))}{7 \cdot 7}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 8.2.1.9.2.3

Rewrite the expression.

$x = \frac{2 (5 - \sqrt{21})}{7}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{2 (5 - \sqrt{21})}{7}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{2 (5 - \sqrt{21})}{7}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{2 (5 - \sqrt{21})}{7}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{2 (5 - \sqrt{21})}{7}$

$y = - \frac{7 + \sqrt{21}}{4}$

$x = \frac{2 (5 - \sqrt{21})}{7}$

$y = - \frac{7 + \sqrt{21}}{4}$

Step 9

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(4, \frac{1}{2})$

$(\frac{2 (5 + \sqrt{21})}{7}, - \frac{7 - \sqrt{21}}{4})$

$(\frac{2 (5 - \sqrt{21})}{7}, - \frac{7 + \sqrt{21}}{4})$

Step 10

The result can be shown in multiple forms.

Point Form:

$(4, \frac{1}{2}), (\frac{2 (5 + \sqrt{21})}{7}, - \frac{7 - \sqrt{21}}{4}), (\frac{2 (5 - \sqrt{21})}{7}, - \frac{7 + \sqrt{21}}{4})$

Equation Form:

$x = 4, y = \frac{1}{2}$

$x = \frac{2 (5 + \sqrt{21})}{7}, y = - \frac{7 - \sqrt{21}}{4}$

$x = \frac{2 (5 - \sqrt{21})}{7}, y = - \frac{7 + \sqrt{21}}{4}$

Step 11