Algebra Examples

$x + y = 11$ $4 x^{2} - 3 y^{2} = 8$

Step 1

Subtract $y$ from both sides of the equation.

$x = 11 - y$

$4 x^{2} - 3 y^{2} = 8$

Step 2

Replace all occurrences of $x$ with $11 - y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $4 x^{2} - 3 y^{2} = 8$ with $11 - y$ .

$4 {(11 - y)}^{2} - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 {(11 - y)}^{2} - 3 y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(11 - y)}^{2}$ as $(11 - y) (11 - y)$ .

$4 ((11 - y) (11 - y)) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.2

Expand $(11 - y) (11 - y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$4 (11 (11 - y) - y (11 - y)) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.2.2

Apply the distributive property.

$4 (11 \cdot 11 + 11 (- y) - y (11 - y)) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.2.3

Apply the distributive property.

$4 (11 \cdot 11 + 11 (- y) - y \cdot 11 - y (- y)) - 3 y^{2} = 8$

$x = 11 - y$

$4 (11 \cdot 11 + 11 (- y) - y \cdot 11 - y (- y)) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.3

Simplify and combine like terms.

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Step 2.2.1.1.3.1

Simplify each term.

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Step 2.2.1.1.3.1.1

Multiply $11$ by $11$ .

$4 (121 + 11 (- y) - y \cdot 11 - y (- y)) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.3.1.2

Multiply $- 1$ by $11$ .

$4 (121 - 11 y - y \cdot 11 - y (- y)) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.3.1.3

Multiply $11$ by $- 1$ .

$4 (121 - 11 y - 11 y - y (- y)) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$4 (121 - 11 y - 11 y - 1 \cdot (- 1 y \cdot y)) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.3.1.5

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

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Step 2.2.1.1.3.1.5.1

Move $y$ .

$4 (121 - 11 y - 11 y - 1 \cdot (- 1 (y \cdot y))) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$4 (121 - 11 y - 11 y - 1 \cdot (- 1 y^{2})) - 3 y^{2} = 8$

$x = 11 - y$

$4 (121 - 11 y - 11 y - 1 \cdot (- 1 y^{2})) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.3.1.6

Multiply $- 1$ by $- 1$ .

$4 (121 - 11 y - 11 y + 1 y^{2}) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.3.1.7

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

$4 (121 - 11 y - 11 y + y^{2}) - 3 y^{2} = 8$

$x = 11 - y$

$4 (121 - 11 y - 11 y + y^{2}) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.3.2

Subtract $11 y$ from $- 11 y$ .

$4 (121 - 22 y + y^{2}) - 3 y^{2} = 8$

$x = 11 - y$

$4 (121 - 22 y + y^{2}) - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.4

Apply the distributive property.

$4 \cdot 121 + 4 (- 22 y) + 4 y^{2} - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.5

Simplify.

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Step 2.2.1.1.5.1

Multiply $4$ by $121$ .

$484 + 4 (- 22 y) + 4 y^{2} - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.1.5.2

Multiply $- 22$ by $4$ .

$484 - 88 y + 4 y^{2} - 3 y^{2} = 8$

$x = 11 - y$

$484 - 88 y + 4 y^{2} - 3 y^{2} = 8$

$x = 11 - y$

$484 - 88 y + 4 y^{2} - 3 y^{2} = 8$

$x = 11 - y$

Step 2.2.1.2

Subtract $3 y^{2}$ from $4 y^{2}$ .

$484 - 88 y + y^{2} = 8$

$x = 11 - y$

$484 - 88 y + y^{2} = 8$

$x = 11 - y$

$484 - 88 y + y^{2} = 8$

$x = 11 - y$

$484 - 88 y + y^{2} = 8$

$x = 11 - y$

Step 3

Solve for $y$ in $484 - 88 y + y^{2} = 8$ .

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Step 3.1

Move all terms to the left side of the equation and simplify.

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Step 3.1.1

Subtract $8$ from both sides of the equation.

$484 - 88 y + y^{2} - 8 = 0$

$x = 11 - y$

Step 3.1.2

Subtract $8$ from $484$ .

$- 88 y + y^{2} + 476 = 0$

$x = 11 - y$

$- 88 y + y^{2} + 476 = 0$

$x = 11 - y$

Step 3.2

Use the quadratic formula to find the solutions.

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

$x = 11 - y$

Step 3.3

Substitute the values $a = 1$ , $b = - 88$ , and $c = 476$ into the quadratic formula and solve for $y$ .

$\frac{88 \pm \sqrt{{(- 88)}^{2} - 4 \cdot (1 \cdot 476)}}{2 \cdot 1}$

$x = 11 - y$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Raise $- 88$ to the power of $2$ .

$y = \frac{88 \pm \sqrt{7744 - 4 \cdot 1 \cdot 476}}{2 \cdot 1}$

$x = 11 - y$

Step 3.4.1.2

Multiply $- 4 \cdot 1 \cdot 476$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{88 \pm \sqrt{7744 - 4 \cdot 476}}{2 \cdot 1}$

$x = 11 - y$

Step 3.4.1.2.2

Multiply $- 4$ by $476$ .

$y = \frac{88 \pm \sqrt{7744 - 1904}}{2 \cdot 1}$

$x = 11 - y$

$y = \frac{88 \pm \sqrt{7744 - 1904}}{2 \cdot 1}$

$x = 11 - y$

Step 3.4.1.3

Subtract $1904$ from $7744$ .

$y = \frac{88 \pm \sqrt{5840}}{2 \cdot 1}$

$x = 11 - y$

Step 3.4.1.4

Rewrite $5840$ as $4^{2} \cdot 365$ .

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Step 3.4.1.4.1

Factor $16$ out of $5840$ .

$y = \frac{88 \pm \sqrt{16 (365)}}{2 \cdot 1}$

$x = 11 - y$

Step 3.4.1.4.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{88 \pm \sqrt{4^{2} \cdot 365}}{2 \cdot 1}$

$x = 11 - y$

$y = \frac{88 \pm \sqrt{4^{2} \cdot 365}}{2 \cdot 1}$

$x = 11 - y$

Step 3.4.1.5

Pull terms out from under the radical.

$y = \frac{88 \pm 4 \sqrt{365}}{2 \cdot 1}$

$x = 11 - y$

$y = \frac{88 \pm 4 \sqrt{365}}{2 \cdot 1}$

$x = 11 - y$

Step 3.4.2

Multiply $2$ by $1$ .

$y = \frac{88 \pm 4 \sqrt{365}}{2}$

$x = 11 - y$

Step 3.4.3

Simplify $\frac{88 \pm 4 \sqrt{365}}{2}$ .

$y = 44 \pm 2 \sqrt{365}$

$x = 11 - y$

$y = 44 \pm 2 \sqrt{365}$

$x = 11 - y$

Step 3.5

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

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $- 88$ to the power of $2$ .

$y = \frac{88 \pm \sqrt{7744 - 4 \cdot 1 \cdot 476}}{2 \cdot 1}$

$x = 11 - y$

Step 3.5.1.2

Multiply $- 4 \cdot 1 \cdot 476$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{88 \pm \sqrt{7744 - 4 \cdot 476}}{2 \cdot 1}$

$x = 11 - y$

Step 3.5.1.2.2

Multiply $- 4$ by $476$ .

$y = \frac{88 \pm \sqrt{7744 - 1904}}{2 \cdot 1}$

$x = 11 - y$

$y = \frac{88 \pm \sqrt{7744 - 1904}}{2 \cdot 1}$

$x = 11 - y$

Step 3.5.1.3

Subtract $1904$ from $7744$ .

$y = \frac{88 \pm \sqrt{5840}}{2 \cdot 1}$

$x = 11 - y$

Step 3.5.1.4

Rewrite $5840$ as $4^{2} \cdot 365$ .

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Step 3.5.1.4.1

Factor $16$ out of $5840$ .

$y = \frac{88 \pm \sqrt{16 (365)}}{2 \cdot 1}$

$x = 11 - y$

Step 3.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{88 \pm \sqrt{4^{2} \cdot 365}}{2 \cdot 1}$

$x = 11 - y$

$y = \frac{88 \pm \sqrt{4^{2} \cdot 365}}{2 \cdot 1}$

$x = 11 - y$

Step 3.5.1.5

Pull terms out from under the radical.

$y = \frac{88 \pm 4 \sqrt{365}}{2 \cdot 1}$

$x = 11 - y$

$y = \frac{88 \pm 4 \sqrt{365}}{2 \cdot 1}$

$x = 11 - y$

Step 3.5.2

Multiply $2$ by $1$ .

$y = \frac{88 \pm 4 \sqrt{365}}{2}$

$x = 11 - y$

Step 3.5.3

Simplify $\frac{88 \pm 4 \sqrt{365}}{2}$ .

$y = 44 \pm 2 \sqrt{365}$

$x = 11 - y$

Step 3.5.4

Change the $\pm$ to $+$ .

$y = 44 + 2 \sqrt{365}$

$x = 11 - y$

$y = 44 + 2 \sqrt{365}$

$x = 11 - y$

Step 3.6

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

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 88$ to the power of $2$ .

$y = \frac{88 \pm \sqrt{7744 - 4 \cdot 1 \cdot 476}}{2 \cdot 1}$

$x = 11 - y$

Step 3.6.1.2

Multiply $- 4 \cdot 1 \cdot 476$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{88 \pm \sqrt{7744 - 4 \cdot 476}}{2 \cdot 1}$

$x = 11 - y$

Step 3.6.1.2.2

Multiply $- 4$ by $476$ .

$y = \frac{88 \pm \sqrt{7744 - 1904}}{2 \cdot 1}$

$x = 11 - y$

$y = \frac{88 \pm \sqrt{7744 - 1904}}{2 \cdot 1}$

$x = 11 - y$

Step 3.6.1.3

Subtract $1904$ from $7744$ .

$y = \frac{88 \pm \sqrt{5840}}{2 \cdot 1}$

$x = 11 - y$

Step 3.6.1.4

Rewrite $5840$ as $4^{2} \cdot 365$ .

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Step 3.6.1.4.1

Factor $16$ out of $5840$ .

$y = \frac{88 \pm \sqrt{16 (365)}}{2 \cdot 1}$

$x = 11 - y$

Step 3.6.1.4.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{88 \pm \sqrt{4^{2} \cdot 365}}{2 \cdot 1}$

$x = 11 - y$

$y = \frac{88 \pm \sqrt{4^{2} \cdot 365}}{2 \cdot 1}$

$x = 11 - y$

Step 3.6.1.5

Pull terms out from under the radical.

$y = \frac{88 \pm 4 \sqrt{365}}{2 \cdot 1}$

$x = 11 - y$

$y = \frac{88 \pm 4 \sqrt{365}}{2 \cdot 1}$

$x = 11 - y$

Step 3.6.2

Multiply $2$ by $1$ .

$y = \frac{88 \pm 4 \sqrt{365}}{2}$

$x = 11 - y$

Step 3.6.3

Simplify $\frac{88 \pm 4 \sqrt{365}}{2}$ .

$y = 44 \pm 2 \sqrt{365}$

$x = 11 - y$

Step 3.6.4

Change the $\pm$ to $-$ .

$y = 44 - 2 \sqrt{365}$

$x = 11 - y$

$y = 44 - 2 \sqrt{365}$

$x = 11 - y$

Step 3.7

The final answer is the combination of both solutions.

$y = 44 + 2 \sqrt{365}, 44 - 2 \sqrt{365}$

$x = 11 - y$

$y = 44 + 2 \sqrt{365}, 44 - 2 \sqrt{365}$

$x = 11 - y$

Step 4

Replace all occurrences of $y$ with $44 + 2 \sqrt{365}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 11 - y$ with $44 + 2 \sqrt{365}$ .

$x = 11 - (44 + 2 \sqrt{365})$

$y = 44 + 2 \sqrt{365}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $11 - (44 + 2 \sqrt{365})$ .

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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.

$x = 11 - 1 \cdot 44 - (2 \sqrt{365})$

$y = 44 + 2 \sqrt{365}$

Step 4.2.1.1.2

Multiply $- 1$ by $44$ .

$x = 11 - 44 - (2 \sqrt{365})$

$y = 44 + 2 \sqrt{365}$

Step 4.2.1.1.3

Multiply $2$ by $- 1$ .

$x = 11 - 44 - 2 \sqrt{365}$

$y = 44 + 2 \sqrt{365}$

$x = 11 - 44 - 2 \sqrt{365}$

$y = 44 + 2 \sqrt{365}$

Step 4.2.1.2

Subtract $44$ from $11$ .

$x = - 33 - 2 \sqrt{365}$

$y = 44 + 2 \sqrt{365}$

$x = - 33 - 2 \sqrt{365}$

$y = 44 + 2 \sqrt{365}$

$x = - 33 - 2 \sqrt{365}$

$y = 44 + 2 \sqrt{365}$

$x = - 33 - 2 \sqrt{365}$

$y = 44 + 2 \sqrt{365}$

Step 5

Replace all occurrences of $y$ with $44 - 2 \sqrt{365}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 11 - y$ with $44 - 2 \sqrt{365}$ .

$x = 11 - (44 - 2 \sqrt{365})$

$y = 44 - 2 \sqrt{365}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $11 - (44 - 2 \sqrt{365})$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$x = 11 - 1 \cdot 44 - (- 2 \sqrt{365})$

$y = 44 - 2 \sqrt{365}$

Step 5.2.1.1.2

Multiply $- 1$ by $44$ .

$x = 11 - 44 - (- 2 \sqrt{365})$

$y = 44 - 2 \sqrt{365}$

Step 5.2.1.1.3

Multiply $- 2$ by $- 1$ .

$x = 11 - 44 + 2 \sqrt{365}$

$y = 44 - 2 \sqrt{365}$

$x = 11 - 44 + 2 \sqrt{365}$

$y = 44 - 2 \sqrt{365}$

Step 5.2.1.2

Subtract $44$ from $11$ .

$x = - 33 + 2 \sqrt{365}$

$y = 44 - 2 \sqrt{365}$

$x = - 33 + 2 \sqrt{365}$

$y = 44 - 2 \sqrt{365}$

$x = - 33 + 2 \sqrt{365}$

$y = 44 - 2 \sqrt{365}$

$x = - 33 + 2 \sqrt{365}$

$y = 44 - 2 \sqrt{365}$

Step 6

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

$(- 33 - 2 \sqrt{365}, 44 + 2 \sqrt{365})$

$(- 33 + 2 \sqrt{365}, 44 - 2 \sqrt{365})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- 33 - 2 \sqrt{365}, 44 + 2 \sqrt{365}), (- 33 + 2 \sqrt{365}, 44 - 2 \sqrt{365})$

Equation Form:

$x = - 33 - 2 \sqrt{365}, y = 44 + 2 \sqrt{365}$

$x = - 33 + 2 \sqrt{365}, y = 44 - 2 \sqrt{365}$

Step 8