Precalculus Examples

$x^{2} - y^{2} = 21$ , $x + y = 7$

Step 1

Subtract $y$ from both sides of the equation.

$x = 7 - y$

$x^{2} - y^{2} = 21$

Step 2

Replace all occurrences of $x$ with $7 - y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} - y^{2} = 21$ with $7 - y$ .

${(7 - y)}^{2} - y^{2} = 21$

$x = 7 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(7 - y)}^{2} - y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(7 - y)}^{2}$ as $(7 - y) (7 - y)$ .

$(7 - y) (7 - y) - y^{2} = 21$

$x = 7 - y$

Step 2.2.1.1.2

Expand $(7 - y) (7 - y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$7 (7 - y) - y (7 - y) - y^{2} = 21$

$x = 7 - y$

Step 2.2.1.1.2.2

Apply the distributive property.

$7 \cdot 7 + 7 (- y) - y (7 - y) - y^{2} = 21$

$x = 7 - y$

Step 2.2.1.1.2.3

Apply the distributive property.

$7 \cdot 7 + 7 (- y) - y \cdot 7 - y (- y) - y^{2} = 21$

$x = 7 - y$

$7 \cdot 7 + 7 (- y) - y \cdot 7 - y (- y) - y^{2} = 21$

$x = 7 - y$

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 $7$ by $7$ .

$49 + 7 (- y) - y \cdot 7 - y (- y) - y^{2} = 21$

$x = 7 - y$

Step 2.2.1.1.3.1.2

Multiply $- 1$ by $7$ .

$49 - 7 y - y \cdot 7 - y (- y) - y^{2} = 21$

$x = 7 - y$

Step 2.2.1.1.3.1.3

Multiply $7$ by $- 1$ .

$49 - 7 y - 7 y - y (- y) - y^{2} = 21$

$x = 7 - y$

Step 2.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$49 - 7 y - 7 y - 1 \cdot (- 1 y \cdot y) - y^{2} = 21$

$x = 7 - y$

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 $y$ .

$49 - 7 y - 7 y - 1 \cdot (- 1 (y \cdot y)) - y^{2} = 21$

$x = 7 - y$

Step 2.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$49 - 7 y - 7 y - 1 \cdot (- 1 y^{2}) - y^{2} = 21$

$x = 7 - y$

$49 - 7 y - 7 y - 1 \cdot (- 1 y^{2}) - y^{2} = 21$

$x = 7 - y$

Step 2.2.1.1.3.1.6

Multiply $- 1$ by $- 1$ .

$49 - 7 y - 7 y + 1 y^{2} - y^{2} = 21$

$x = 7 - y$

Step 2.2.1.1.3.1.7

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

$49 - 7 y - 7 y + y^{2} - y^{2} = 21$

$x = 7 - y$

$49 - 7 y - 7 y + y^{2} - y^{2} = 21$

$x = 7 - y$

Step 2.2.1.1.3.2

Subtract $7 y$ from $- 7 y$ .

$49 - 14 y + y^{2} - y^{2} = 21$

$x = 7 - y$

$49 - 14 y + y^{2} - y^{2} = 21$

$x = 7 - y$

$49 - 14 y + y^{2} - y^{2} = 21$

$x = 7 - y$

Step 2.2.1.2

Combine the opposite terms in $49 - 14 y + y^{2} - y^{2}$ .

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Step 2.2.1.2.1

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

$49 - 14 y + 0 = 21$

$x = 7 - y$

Step 2.2.1.2.2

Add $49 - 14 y$ and $0$ .

$49 - 14 y = 21$

$x = 7 - y$

$49 - 14 y = 21$

$x = 7 - y$

$49 - 14 y = 21$

$x = 7 - y$

$49 - 14 y = 21$

$x = 7 - y$

$49 - 14 y = 21$

$x = 7 - y$

Step 3

Solve for $y$ in $49 - 14 y = 21$ .

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Step 3.1

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

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Step 3.1.1

Subtract $49$ from both sides of the equation.

$- 14 y = 21 - 49$

$x = 7 - y$

Step 3.1.2

Subtract $49$ from $21$ .

$- 14 y = - 28$

$x = 7 - y$

$- 14 y = - 28$

$x = 7 - y$

Step 3.2

Divide each term in $- 14 y = - 28$ by $- 14$ and simplify.

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Step 3.2.1

Divide each term in $- 14 y = - 28$ by $- 14$ .

$\frac{- 14 y}{- 14} = \frac{- 28}{- 14}$

$x = 7 - y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 14$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 14 y}{- 14} = \frac{- 28}{- 14}$

$x = 7 - y$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 28}{- 14}$

$x = 7 - y$

$y = \frac{- 28}{- 14}$

$x = 7 - y$

$y = \frac{- 28}{- 14}$

$x = 7 - y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 28$ by $- 14$ .

$y = 2$

$x = 7 - y$

$y = 2$

$x = 7 - y$

$y = 2$

$x = 7 - y$

$y = 2$

$x = 7 - y$

Step 4

Replace all occurrences of $y$ with $2$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 7 - y$ with $2$ .

$x = 7 - (2)$

$y = 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $7 - (2)$ .

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Step 4.2.1.1

Multiply $- 1$ by $2$ .

$x = 7 - 2$

$y = 2$

Step 4.2.1.2

Subtract $2$ from $7$ .

$x = 5$

$y = 2$

$x = 5$

$y = 2$

$x = 5$

$y = 2$

$x = 5$

$y = 2$

Step 5

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

$(5, 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(5, 2)$

Equation Form:

$x = 5, y = 2$

Step 7