Algebra Examples

$x + y = 50$ $x y = 300$

Step 1

Subtract $y$ from both sides of the equation.

$x = 50 - y$

$x y = 300$

Step 2

Replace all occurrences of $x$ with $50 - y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x y = 300$ with $50 - y$ .

$(50 - y) \cdot y = 300$

$x = 50 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $(50 - y) \cdot y$ .

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Step 2.2.1.1

Apply the distributive property.

$50 y - y \cdot y = 300$

$x = 50 - y$

Step 2.2.1.2

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

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Step 2.2.1.2.1

Move $y$ .

$50 y - (y \cdot y) = 300$

$x = 50 - y$

Step 2.2.1.2.2

Multiply $y$ by $y$ .

$50 y - y^{2} = 300$

$x = 50 - y$

$50 y - y^{2} = 300$

$x = 50 - y$

$50 y - y^{2} = 300$

$x = 50 - y$

$50 y - y^{2} = 300$

$x = 50 - y$

$50 y - y^{2} = 300$

$x = 50 - y$

Step 3

Solve for $y$ in $50 y - y^{2} = 300$ .

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Step 3.1

Subtract $300$ from both sides of the equation.

$50 y - y^{2} - 300 = 0$

$x = 50 - y$

Step 3.2

Use the quadratic formula to find the solutions.

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

$x = 50 - y$

Step 3.3

Substitute the values $a = - 1$ , $b = 50$ , and $c = - 300$ into the quadratic formula and solve for $y$ .

$\frac{- 50 \pm \sqrt{50^{2} - 4 \cdot (- 1 \cdot - 300)}}{2 \cdot - 1}$

$x = 50 - y$

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 $50$ to the power of $2$ .

$y = \frac{- 50 \pm \sqrt{2500 - 4 \cdot - 1 \cdot - 300}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.4.1.2

Multiply $- 4 \cdot - 1 \cdot - 300$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $- 1$ .

$y = \frac{- 50 \pm \sqrt{2500 + 4 \cdot - 300}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.4.1.2.2

Multiply $4$ by $- 300$ .

$y = \frac{- 50 \pm \sqrt{2500 - 1200}}{2 \cdot - 1}$

$x = 50 - y$

$y = \frac{- 50 \pm \sqrt{2500 - 1200}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.4.1.3

Subtract $1200$ from $2500$ .

$y = \frac{- 50 \pm \sqrt{1300}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.4.1.4

Rewrite $1300$ as $10^{2} \cdot 13$ .

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Step 3.4.1.4.1

Factor $100$ out of $1300$ .

$y = \frac{- 50 \pm \sqrt{100 (13)}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.4.1.4.2

Rewrite $100$ as $10^{2}$ .

$y = \frac{- 50 \pm \sqrt{10^{2} \cdot 13}}{2 \cdot - 1}$

$x = 50 - y$

$y = \frac{- 50 \pm \sqrt{10^{2} \cdot 13}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.4.1.5

Pull terms out from under the radical.

$y = \frac{- 50 \pm 10 \sqrt{13}}{2 \cdot - 1}$

$x = 50 - y$

$y = \frac{- 50 \pm 10 \sqrt{13}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.4.2

Multiply $2$ by $- 1$ .

$y = \frac{- 50 \pm 10 \sqrt{13}}{- 2}$

$x = 50 - y$

Step 3.4.3

Simplify $\frac{- 50 \pm 10 \sqrt{13}}{- 2}$ .

$y = 25 \pm 5 \sqrt{13}$

$x = 50 - y$

$y = 25 \pm 5 \sqrt{13}$

$x = 50 - y$

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 $50$ to the power of $2$ .

$y = \frac{- 50 \pm \sqrt{2500 - 4 \cdot - 1 \cdot - 300}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.5.1.2

Multiply $- 4 \cdot - 1 \cdot - 300$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $- 1$ .

$y = \frac{- 50 \pm \sqrt{2500 + 4 \cdot - 300}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.5.1.2.2

Multiply $4$ by $- 300$ .

$y = \frac{- 50 \pm \sqrt{2500 - 1200}}{2 \cdot - 1}$

$x = 50 - y$

$y = \frac{- 50 \pm \sqrt{2500 - 1200}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.5.1.3

Subtract $1200$ from $2500$ .

$y = \frac{- 50 \pm \sqrt{1300}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.5.1.4

Rewrite $1300$ as $10^{2} \cdot 13$ .

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Step 3.5.1.4.1

Factor $100$ out of $1300$ .

$y = \frac{- 50 \pm \sqrt{100 (13)}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.5.1.4.2

Rewrite $100$ as $10^{2}$ .

$y = \frac{- 50 \pm \sqrt{10^{2} \cdot 13}}{2 \cdot - 1}$

$x = 50 - y$

$y = \frac{- 50 \pm \sqrt{10^{2} \cdot 13}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.5.1.5

Pull terms out from under the radical.

$y = \frac{- 50 \pm 10 \sqrt{13}}{2 \cdot - 1}$

$x = 50 - y$

$y = \frac{- 50 \pm 10 \sqrt{13}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.5.2

Multiply $2$ by $- 1$ .

$y = \frac{- 50 \pm 10 \sqrt{13}}{- 2}$

$x = 50 - y$

Step 3.5.3

Simplify $\frac{- 50 \pm 10 \sqrt{13}}{- 2}$ .

$y = 25 \pm 5 \sqrt{13}$

$x = 50 - y$

Step 3.5.4

Change the $\pm$ to $+$ .

$y = 25 + 5 \sqrt{13}$

$x = 50 - y$

$y = 25 + 5 \sqrt{13}$

$x = 50 - y$

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 $50$ to the power of $2$ .

$y = \frac{- 50 \pm \sqrt{2500 - 4 \cdot - 1 \cdot - 300}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.6.1.2

Multiply $- 4 \cdot - 1 \cdot - 300$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $- 1$ .

$y = \frac{- 50 \pm \sqrt{2500 + 4 \cdot - 300}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.6.1.2.2

Multiply $4$ by $- 300$ .

$y = \frac{- 50 \pm \sqrt{2500 - 1200}}{2 \cdot - 1}$

$x = 50 - y$

$y = \frac{- 50 \pm \sqrt{2500 - 1200}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.6.1.3

Subtract $1200$ from $2500$ .

$y = \frac{- 50 \pm \sqrt{1300}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.6.1.4

Rewrite $1300$ as $10^{2} \cdot 13$ .

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Step 3.6.1.4.1

Factor $100$ out of $1300$ .

$y = \frac{- 50 \pm \sqrt{100 (13)}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.6.1.4.2

Rewrite $100$ as $10^{2}$ .

$y = \frac{- 50 \pm \sqrt{10^{2} \cdot 13}}{2 \cdot - 1}$

$x = 50 - y$

$y = \frac{- 50 \pm \sqrt{10^{2} \cdot 13}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.6.1.5

Pull terms out from under the radical.

$y = \frac{- 50 \pm 10 \sqrt{13}}{2 \cdot - 1}$

$x = 50 - y$

$y = \frac{- 50 \pm 10 \sqrt{13}}{2 \cdot - 1}$

$x = 50 - y$

Step 3.6.2

Multiply $2$ by $- 1$ .

$y = \frac{- 50 \pm 10 \sqrt{13}}{- 2}$

$x = 50 - y$

Step 3.6.3

Simplify $\frac{- 50 \pm 10 \sqrt{13}}{- 2}$ .

$y = 25 \pm 5 \sqrt{13}$

$x = 50 - y$

Step 3.6.4

Change the $\pm$ to $-$ .

$y = 25 - 5 \sqrt{13}$

$x = 50 - y$

$y = 25 - 5 \sqrt{13}$

$x = 50 - y$

Step 3.7

The final answer is the combination of both solutions.

$y = 25 + 5 \sqrt{13}, 25 - 5 \sqrt{13}$

$x = 50 - y$

$y = 25 + 5 \sqrt{13}, 25 - 5 \sqrt{13}$

$x = 50 - y$

Step 4

Replace all occurrences of $y$ with $25 + 5 \sqrt{13}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 50 - y$ with $25 + 5 \sqrt{13}$ .

$x = 50 - (25 + 5 \sqrt{13})$

$y = 25 + 5 \sqrt{13}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $50 - (25 + 5 \sqrt{13})$ .

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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.

$x = 50 - 1 \cdot 25 - (5 \sqrt{13})$

$y = 25 + 5 \sqrt{13}$

Step 4.2.1.1.2

Multiply $- 1$ by $25$ .

$x = 50 - 25 - (5 \sqrt{13})$

$y = 25 + 5 \sqrt{13}$

Step 4.2.1.1.3

Multiply $5$ by $- 1$ .

$x = 50 - 25 - 5 \sqrt{13}$

$y = 25 + 5 \sqrt{13}$

$x = 50 - 25 - 5 \sqrt{13}$

$y = 25 + 5 \sqrt{13}$

Step 4.2.1.2

Subtract $25$ from $50$ .

$x = 25 - 5 \sqrt{13}$

$y = 25 + 5 \sqrt{13}$

$x = 25 - 5 \sqrt{13}$

$y = 25 + 5 \sqrt{13}$

$x = 25 - 5 \sqrt{13}$

$y = 25 + 5 \sqrt{13}$

$x = 25 - 5 \sqrt{13}$

$y = 25 + 5 \sqrt{13}$

Step 5

Replace all occurrences of $y$ with $25 - 5 \sqrt{13}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 50 - y$ with $25 - 5 \sqrt{13}$ .

$x = 50 - (25 - 5 \sqrt{13})$

$y = 25 - 5 \sqrt{13}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $50 - (25 - 5 \sqrt{13})$ .

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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.

$x = 50 - 1 \cdot 25 - (- 5 \sqrt{13})$

$y = 25 - 5 \sqrt{13}$

Step 5.2.1.1.2

Multiply $- 1$ by $25$ .

$x = 50 - 25 - (- 5 \sqrt{13})$

$y = 25 - 5 \sqrt{13}$

Step 5.2.1.1.3

Multiply $- 5$ by $- 1$ .

$x = 50 - 25 + 5 \sqrt{13}$

$y = 25 - 5 \sqrt{13}$

$x = 50 - 25 + 5 \sqrt{13}$

$y = 25 - 5 \sqrt{13}$

Step 5.2.1.2

Subtract $25$ from $50$ .

$x = 25 + 5 \sqrt{13}$

$y = 25 - 5 \sqrt{13}$

$x = 25 + 5 \sqrt{13}$

$y = 25 - 5 \sqrt{13}$

$x = 25 + 5 \sqrt{13}$

$y = 25 - 5 \sqrt{13}$

$x = 25 + 5 \sqrt{13}$

$y = 25 - 5 \sqrt{13}$

Step 6

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

$(25 - 5 \sqrt{13}, 25 + 5 \sqrt{13})$

$(25 + 5 \sqrt{13}, 25 - 5 \sqrt{13})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(25 - 5 \sqrt{13}, 25 + 5 \sqrt{13}), (25 + 5 \sqrt{13}, 25 - 5 \sqrt{13})$

Equation Form:

$x = 25 - 5 \sqrt{13}, y = 25 + 5 \sqrt{13}$

$x = 25 + 5 \sqrt{13}, y = 25 - 5 \sqrt{13}$

Step 8