Algebra Examples

$7 x^{2} + 3 y^{2} - 24 y = 139$ $x - y = 9$

Step 1

Add $y$ to both sides of the equation.

$x = 9 + y$

$7 x^{2} + 3 y^{2} - 24 y = 139$

Step 2

Replace all occurrences of $x$ with $9 + y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $7 x^{2} + 3 y^{2} - 24 y = 139$ with $9 + y$ .

$7 {(9 + y)}^{2} + 3 y^{2} - 24 y = 139$

$x = 9 + y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $7 {(9 + y)}^{2} + 3 y^{2} - 24 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(9 + y)}^{2}$ as $(9 + y) (9 + y)$ .

$7 ((9 + y) (9 + y)) + 3 y^{2} - 24 y = 139$

$x = 9 + y$

Step 2.2.1.1.2

Expand $(9 + y) (9 + y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$7 (9 (9 + y) + y (9 + y)) + 3 y^{2} - 24 y = 139$

$x = 9 + y$

Step 2.2.1.1.2.2

Apply the distributive property.

$7 (9 \cdot 9 + 9 y + y (9 + y)) + 3 y^{2} - 24 y = 139$

$x = 9 + y$

Step 2.2.1.1.2.3

Apply the distributive property.

$7 (9 \cdot 9 + 9 y + y \cdot 9 + y \cdot y) + 3 y^{2} - 24 y = 139$

$x = 9 + y$

$7 (9 \cdot 9 + 9 y + y \cdot 9 + y \cdot y) + 3 y^{2} - 24 y = 139$

$x = 9 + y$

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 $9$ by $9$ .

$7 (81 + 9 y + y \cdot 9 + y \cdot y) + 3 y^{2} - 24 y = 139$

$x = 9 + y$

Step 2.2.1.1.3.1.2

Move $9$ to the left of $y$ .

$7 (81 + 9 y + 9 \cdot y + y \cdot y) + 3 y^{2} - 24 y = 139$

$x = 9 + y$

Step 2.2.1.1.3.1.3

Multiply $y$ by $y$ .

$7 (81 + 9 y + 9 y + y^{2}) + 3 y^{2} - 24 y = 139$

$x = 9 + y$

$7 (81 + 9 y + 9 y + y^{2}) + 3 y^{2} - 24 y = 139$

$x = 9 + y$

Step 2.2.1.1.3.2

Add $9 y$ and $9 y$ .

$7 (81 + 18 y + y^{2}) + 3 y^{2} - 24 y = 139$

$x = 9 + y$

$7 (81 + 18 y + y^{2}) + 3 y^{2} - 24 y = 139$

$x = 9 + y$

Step 2.2.1.1.4

Apply the distributive property.

$7 \cdot 81 + 7 (18 y) + 7 y^{2} + 3 y^{2} - 24 y = 139$

$x = 9 + y$

Step 2.2.1.1.5

Simplify.

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Step 2.2.1.1.5.1

Multiply $7$ by $81$ .

$567 + 7 (18 y) + 7 y^{2} + 3 y^{2} - 24 y = 139$

$x = 9 + y$

Step 2.2.1.1.5.2

Multiply $18$ by $7$ .

$567 + 126 y + 7 y^{2} + 3 y^{2} - 24 y = 139$

$x = 9 + y$

$567 + 126 y + 7 y^{2} + 3 y^{2} - 24 y = 139$

$x = 9 + y$

$567 + 126 y + 7 y^{2} + 3 y^{2} - 24 y = 139$

$x = 9 + y$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $24 y$ from $126 y$ .

$567 + 7 y^{2} + 3 y^{2} + 102 y = 139$

$x = 9 + y$

Step 2.2.1.2.2

Add $7 y^{2}$ and $3 y^{2}$ .

$567 + 10 y^{2} + 102 y = 139$

$x = 9 + y$

$567 + 10 y^{2} + 102 y = 139$

$x = 9 + y$

$567 + 10 y^{2} + 102 y = 139$

$x = 9 + y$

$567 + 10 y^{2} + 102 y = 139$

$x = 9 + y$

$567 + 10 y^{2} + 102 y = 139$

$x = 9 + y$

Step 3

Solve for $y$ in $567 + 10 y^{2} + 102 y = 139$ .

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Step 3.1

Subtract $139$ from both sides of the equation.

$567 + 10 y^{2} + 102 y - 139 = 0$

$x = 9 + y$

Step 3.2

Subtract $139$ from $567$ .

$10 y^{2} + 102 y + 428 = 0$

$x = 9 + y$

Step 3.3

Factor $2$ out of $10 y^{2} + 102 y + 428$ .

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Step 3.3.1

Factor $2$ out of $10 y^{2}$ .

$2 (5 y^{2}) + 102 y + 428 = 0$

$x = 9 + y$

Step 3.3.2

Factor $2$ out of $102 y$ .

$2 (5 y^{2}) + 2 (51 y) + 428 = 0$

$x = 9 + y$

Step 3.3.3

Factor $2$ out of $428$ .

$2 (5 y^{2}) + 2 (51 y) + 2 (214) = 0$

$x = 9 + y$

Step 3.3.4

Factor $2$ out of $2 (5 y^{2}) + 2 (51 y)$ .

$2 (5 y^{2} + 51 y) + 2 (214) = 0$

$x = 9 + y$

Step 3.3.5

Factor $2$ out of $2 (5 y^{2} + 51 y) + 2 (214)$ .

$2 (5 y^{2} + 51 y + 214) = 0$

$x = 9 + y$

$2 (5 y^{2} + 51 y + 214) = 0$

$x = 9 + y$

Step 3.4

Divide each term in $2 (5 y^{2} + 51 y + 214) = 0$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 (5 y^{2} + 51 y + 214) = 0$ by $2$ .

$\frac{2 (5 y^{2} + 51 y + 214)}{2} = \frac{0}{2}$

$x = 9 + y$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 (5 y^{2} + 51 y + 214)}{2} = \frac{0}{2}$

$x = 9 + y$

Step 3.4.2.1.2

Divide $5 y^{2} + 51 y + 214$ by $1$ .

$5 y^{2} + 51 y + 214 = \frac{0}{2}$

$x = 9 + y$

$5 y^{2} + 51 y + 214 = \frac{0}{2}$

$x = 9 + y$

$5 y^{2} + 51 y + 214 = \frac{0}{2}$

$x = 9 + y$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $0$ by $2$ .

$5 y^{2} + 51 y + 214 = 0$

$x = 9 + y$

$5 y^{2} + 51 y + 214 = 0$

$x = 9 + y$

$5 y^{2} + 51 y + 214 = 0$

$x = 9 + y$

Step 3.5

Use the quadratic formula to find the solutions.

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

$x = 9 + y$

Step 3.6

Substitute the values $a = 5$ , $b = 51$ , and $c = 214$ into the quadratic formula and solve for $y$ .

$\frac{- 51 \pm \sqrt{51^{2} - 4 \cdot (5 \cdot 214)}}{2 \cdot 5}$

$x = 9 + y$

Step 3.7

Simplify.

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $51$ to the power of $2$ .

$y = \frac{- 51 \pm \sqrt{2601 - 4 \cdot 5 \cdot 214}}{2 \cdot 5}$

$x = 9 + y$

Step 3.7.1.2

Multiply $- 4 \cdot 5 \cdot 214$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $5$ .

$y = \frac{- 51 \pm \sqrt{2601 - 20 \cdot 214}}{2 \cdot 5}$

$x = 9 + y$

Step 3.7.1.2.2

Multiply $- 20$ by $214$ .

$y = \frac{- 51 \pm \sqrt{2601 - 4280}}{2 \cdot 5}$

$x = 9 + y$

$y = \frac{- 51 \pm \sqrt{2601 - 4280}}{2 \cdot 5}$

$x = 9 + y$

Step 3.7.1.3

Subtract $4280$ from $2601$ .

$y = \frac{- 51 \pm \sqrt{- 1679}}{2 \cdot 5}$

$x = 9 + y$

Step 3.7.1.4

Rewrite $- 1679$ as $- 1 (1679)$ .

$y = \frac{- 51 \pm \sqrt{- 1 \cdot 1679}}{2 \cdot 5}$

$x = 9 + y$

Step 3.7.1.5

Rewrite $\sqrt{- 1 (1679)}$ as $\sqrt{- 1} \cdot \sqrt{1679}$ .

$y = \frac{- 51 \pm \sqrt{- 1} \cdot \sqrt{1679}}{2 \cdot 5}$

$x = 9 + y$

Step 3.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 51 \pm i \sqrt{1679}}{2 \cdot 5}$

$x = 9 + y$

$y = \frac{- 51 \pm i \sqrt{1679}}{2 \cdot 5}$

$x = 9 + y$

Step 3.7.2

Multiply $2$ by $5$ .

$y = \frac{- 51 \pm i \sqrt{1679}}{10}$

$x = 9 + y$

$y = \frac{- 51 \pm i \sqrt{1679}}{10}$

$x = 9 + y$

Step 3.8

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

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

Raise $51$ to the power of $2$ .

$y = \frac{- 51 \pm \sqrt{2601 - 4 \cdot 5 \cdot 214}}{2 \cdot 5}$

$x = 9 + y$

Step 3.8.1.2

Multiply $- 4 \cdot 5 \cdot 214$ .

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Step 3.8.1.2.1

Multiply $- 4$ by $5$ .

$y = \frac{- 51 \pm \sqrt{2601 - 20 \cdot 214}}{2 \cdot 5}$

$x = 9 + y$

Step 3.8.1.2.2

Multiply $- 20$ by $214$ .

$y = \frac{- 51 \pm \sqrt{2601 - 4280}}{2 \cdot 5}$

$x = 9 + y$

$y = \frac{- 51 \pm \sqrt{2601 - 4280}}{2 \cdot 5}$

$x = 9 + y$

Step 3.8.1.3

Subtract $4280$ from $2601$ .

$y = \frac{- 51 \pm \sqrt{- 1679}}{2 \cdot 5}$

$x = 9 + y$

Step 3.8.1.4

Rewrite $- 1679$ as $- 1 (1679)$ .

$y = \frac{- 51 \pm \sqrt{- 1 \cdot 1679}}{2 \cdot 5}$

$x = 9 + y$

Step 3.8.1.5

Rewrite $\sqrt{- 1 (1679)}$ as $\sqrt{- 1} \cdot \sqrt{1679}$ .

$y = \frac{- 51 \pm \sqrt{- 1} \cdot \sqrt{1679}}{2 \cdot 5}$

$x = 9 + y$

Step 3.8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 51 \pm i \sqrt{1679}}{2 \cdot 5}$

$x = 9 + y$

$y = \frac{- 51 \pm i \sqrt{1679}}{2 \cdot 5}$

$x = 9 + y$

Step 3.8.2

Multiply $2$ by $5$ .

$y = \frac{- 51 \pm i \sqrt{1679}}{10}$

$x = 9 + y$

Step 3.8.3

Change the $\pm$ to $+$ .

$y = \frac{- 51 + i \sqrt{1679}}{10}$

$x = 9 + y$

Step 3.8.4

Rewrite $- 51$ as $- 1 (51)$ .

$y = \frac{- 1 \cdot 51 + i \sqrt{1679}}{10}$

$x = 9 + y$

Step 3.8.5

Factor $- 1$ out of $i \sqrt{1679}$ .

$y = \frac{- 1 \cdot 51 - (- i \sqrt{1679})}{10}$

$x = 9 + y$

Step 3.8.6

Factor $- 1$ out of $- 1 (51) - (- i \sqrt{1679})$ .

$y = \frac{- 1 (51 - i \sqrt{1679})}{10}$

$x = 9 + y$

Step 3.8.7

Move the negative in front of the fraction.

$y = - \frac{51 - i \sqrt{1679}}{10}$

$x = 9 + y$

$y = - \frac{51 - i \sqrt{1679}}{10}$

$x = 9 + y$

Step 3.9

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

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Step 3.9.1

Simplify the numerator.

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Step 3.9.1.1

Raise $51$ to the power of $2$ .

$y = \frac{- 51 \pm \sqrt{2601 - 4 \cdot 5 \cdot 214}}{2 \cdot 5}$

$x = 9 + y$

Step 3.9.1.2

Multiply $- 4 \cdot 5 \cdot 214$ .

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Step 3.9.1.2.1

Multiply $- 4$ by $5$ .

$y = \frac{- 51 \pm \sqrt{2601 - 20 \cdot 214}}{2 \cdot 5}$

$x = 9 + y$

Step 3.9.1.2.2

Multiply $- 20$ by $214$ .

$y = \frac{- 51 \pm \sqrt{2601 - 4280}}{2 \cdot 5}$

$x = 9 + y$

$y = \frac{- 51 \pm \sqrt{2601 - 4280}}{2 \cdot 5}$

$x = 9 + y$

Step 3.9.1.3

Subtract $4280$ from $2601$ .

$y = \frac{- 51 \pm \sqrt{- 1679}}{2 \cdot 5}$

$x = 9 + y$

Step 3.9.1.4

Rewrite $- 1679$ as $- 1 (1679)$ .

$y = \frac{- 51 \pm \sqrt{- 1 \cdot 1679}}{2 \cdot 5}$

$x = 9 + y$

Step 3.9.1.5

Rewrite $\sqrt{- 1 (1679)}$ as $\sqrt{- 1} \cdot \sqrt{1679}$ .

$y = \frac{- 51 \pm \sqrt{- 1} \cdot \sqrt{1679}}{2 \cdot 5}$

$x = 9 + y$

Step 3.9.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 51 \pm i \sqrt{1679}}{2 \cdot 5}$

$x = 9 + y$

$y = \frac{- 51 \pm i \sqrt{1679}}{2 \cdot 5}$

$x = 9 + y$

Step 3.9.2

Multiply $2$ by $5$ .

$y = \frac{- 51 \pm i \sqrt{1679}}{10}$

$x = 9 + y$

Step 3.9.3

Change the $\pm$ to $-$ .

$y = \frac{- 51 - i \sqrt{1679}}{10}$

$x = 9 + y$

Step 3.9.4

Rewrite $- 51$ as $- 1 (51)$ .

$y = \frac{- 1 \cdot 51 - i \sqrt{1679}}{10}$

$x = 9 + y$

Step 3.9.5

Factor $- 1$ out of $- i \sqrt{1679}$ .

$y = \frac{- 1 \cdot 51 - (i \sqrt{1679})}{10}$

$x = 9 + y$

Step 3.9.6

Factor $- 1$ out of $- 1 (51) - (i \sqrt{1679})$ .

$y = \frac{- 1 (51 + i \sqrt{1679})}{10}$

$x = 9 + y$

Step 3.9.7

Move the negative in front of the fraction.

$y = - \frac{51 + i \sqrt{1679}}{10}$

$x = 9 + y$

$y = - \frac{51 + i \sqrt{1679}}{10}$

$x = 9 + y$

Step 3.10

The final answer is the combination of both solutions.

$y = - \frac{51 - i \sqrt{1679}}{10}, - \frac{51 + i \sqrt{1679}}{10}$

$x = 9 + y$

$y = - \frac{51 - i \sqrt{1679}}{10}, - \frac{51 + i \sqrt{1679}}{10}$

$x = 9 + y$

Step 4

Replace all occurrences of $y$ with $- \frac{51 - i \sqrt{1679}}{10}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 9 + y$ with $- \frac{51 - i \sqrt{1679}}{10}$ .

$x = 9 - \frac{51 - i \sqrt{1679}}{10}$

$y = - \frac{51 - i \sqrt{1679}}{10}$

Step 4.2

Simplify $x = 9 - \frac{51 - i \sqrt{1679}}{10}$ .

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Remove parentheses.

$x = 9 - \frac{51 - i \sqrt{1679}}{10}$

$y = - \frac{51 - i \sqrt{1679}}{10}$

$x = 9 - \frac{51 - i \sqrt{1679}}{10}$

$y = - \frac{51 - i \sqrt{1679}}{10}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $9 - \frac{51 - i \sqrt{1679}}{10}$ .

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Step 4.2.2.1.1

To write $9$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$x = 9 \cdot \frac{10}{10} - \frac{51 - i \sqrt{1679}}{10}$

$y = - \frac{51 - i \sqrt{1679}}{10}$

Step 4.2.2.1.2

Combine $9$ and $\frac{10}{10}$ .

$x = \frac{9 \cdot 10}{10} - \frac{51 - i \sqrt{1679}}{10}$

$y = - \frac{51 - i \sqrt{1679}}{10}$

Step 4.2.2.1.3

Combine the numerators over the common denominator.

$x = \frac{39 + \sqrt{1679} i}{10}$

$y = - \frac{51 - i \sqrt{1679}}{10}$

$x = \frac{39 + \sqrt{1679} i}{10}$

$y = - \frac{51 - i \sqrt{1679}}{10}$

$x = \frac{39 + \sqrt{1679} i}{10}$

$y = - \frac{51 - i \sqrt{1679}}{10}$

$x = \frac{39 + \sqrt{1679} i}{10}$

$y = - \frac{51 - i \sqrt{1679}}{10}$

$x = \frac{39 + \sqrt{1679} i}{10}$

$y = - \frac{51 - i \sqrt{1679}}{10}$

Step 5

Replace all occurrences of $y$ with $- \frac{51 + i \sqrt{1679}}{10}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 9 + y$ with $- \frac{51 + i \sqrt{1679}}{10}$ .

$x = 9 - \frac{51 + i \sqrt{1679}}{10}$

$y = - \frac{51 + i \sqrt{1679}}{10}$

Step 5.2

Simplify $x = 9 - \frac{51 + i \sqrt{1679}}{10}$ .

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Remove parentheses.

$x = 9 - \frac{51 + i \sqrt{1679}}{10}$

$y = - \frac{51 + i \sqrt{1679}}{10}$

$x = 9 - \frac{51 + i \sqrt{1679}}{10}$

$y = - \frac{51 + i \sqrt{1679}}{10}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $9 - \frac{51 + i \sqrt{1679}}{10}$ .

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Step 5.2.2.1.1

To write $9$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$x = 9 \cdot \frac{10}{10} - \frac{51 + i \sqrt{1679}}{10}$

$y = - \frac{51 + i \sqrt{1679}}{10}$

Step 5.2.2.1.2

Combine $9$ and $\frac{10}{10}$ .

$x = \frac{9 \cdot 10}{10} - \frac{51 + i \sqrt{1679}}{10}$

$y = - \frac{51 + i \sqrt{1679}}{10}$

Step 5.2.2.1.3

Combine the numerators over the common denominator.

$x = \frac{39 - i \sqrt{1679}}{10}$

$y = - \frac{51 + i \sqrt{1679}}{10}$

$x = \frac{39 - i \sqrt{1679}}{10}$

$y = - \frac{51 + i \sqrt{1679}}{10}$

$x = \frac{39 - i \sqrt{1679}}{10}$

$y = - \frac{51 + i \sqrt{1679}}{10}$

$x = \frac{39 - i \sqrt{1679}}{10}$

$y = - \frac{51 + i \sqrt{1679}}{10}$

$x = \frac{39 - i \sqrt{1679}}{10}$

$y = - \frac{51 + i \sqrt{1679}}{10}$

Step 6

List all of the solutions.

$x = \frac{39 + \sqrt{1679} i}{10}, y = - \frac{51 - i \sqrt{1679}}{10}$

$x = \frac{39 - i \sqrt{1679}}{10}, y = - \frac{51 + i \sqrt{1679}}{10}$

Step 7