Precalculus Examples

$x^{2} - y^{2} = 9$ , $x - y = 1$

Step 1

Add $y$ to both sides of the equation.

$x = 1 + y$

$x^{2} - y^{2} = 9$

Step 2

Replace all occurrences of $x$ with $1 + y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} - y^{2} = 9$ with $1 + y$ .

${(1 + y)}^{2} - y^{2} = 9$

$x = 1 + y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(1 + y)}^{2} - y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(1 + y)}^{2}$ as $(1 + y) (1 + y)$ .

$(1 + y) (1 + y) - y^{2} = 9$

$x = 1 + y$

Step 2.2.1.1.2

Expand $(1 + y) (1 + y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$1 (1 + y) + y (1 + y) - y^{2} = 9$

$x = 1 + y$

Step 2.2.1.1.2.2

Apply the distributive property.

$1 \cdot 1 + 1 y + y (1 + y) - y^{2} = 9$

$x = 1 + y$

Step 2.2.1.1.2.3

Apply the distributive property.

$1 \cdot 1 + 1 y + y \cdot 1 + y \cdot y - y^{2} = 9$

$x = 1 + y$

$1 \cdot 1 + 1 y + y \cdot 1 + y \cdot y - y^{2} = 9$

$x = 1 + y$

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 $1$ by $1$ .

$1 + 1 y + y \cdot 1 + y \cdot y - y^{2} = 9$

$x = 1 + y$

Step 2.2.1.1.3.1.2

Multiply $y$ by $1$ .

$1 + y + y \cdot 1 + y \cdot y - y^{2} = 9$

$x = 1 + y$

Step 2.2.1.1.3.1.3

Multiply $y$ by $1$ .

$1 + y + y + y \cdot y - y^{2} = 9$

$x = 1 + y$

Step 2.2.1.1.3.1.4

Multiply $y$ by $y$ .

$1 + y + y + y^{2} - y^{2} = 9$

$x = 1 + y$

$1 + y + y + y^{2} - y^{2} = 9$

$x = 1 + y$

Step 2.2.1.1.3.2

Add $y$ and $y$ .

$1 + 2 y + y^{2} - y^{2} = 9$

$x = 1 + y$

$1 + 2 y + y^{2} - y^{2} = 9$

$x = 1 + y$

$1 + 2 y + y^{2} - y^{2} = 9$

$x = 1 + y$

Step 2.2.1.2

Combine the opposite terms in $1 + 2 y + y^{2} - y^{2}$ .

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Step 2.2.1.2.1

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

$1 + 2 y + 0 = 9$

$x = 1 + y$

Step 2.2.1.2.2

Add $1 + 2 y$ and $0$ .

$1 + 2 y = 9$

$x = 1 + y$

$1 + 2 y = 9$

$x = 1 + y$

$1 + 2 y = 9$

$x = 1 + y$

$1 + 2 y = 9$

$x = 1 + y$

$1 + 2 y = 9$

$x = 1 + y$

Step 3

Solve for $y$ in $1 + 2 y = 9$ .

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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 $1$ from both sides of the equation.

$2 y = 9 - 1$

$x = 1 + y$

Step 3.1.2

Subtract $1$ from $9$ .

$2 y = 8$

$x = 1 + y$

$2 y = 8$

$x = 1 + y$

Step 3.2

Divide each term in $2 y = 8$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 y = 8$ by $2$ .

$\frac{2 y}{2} = \frac{8}{2}$

$x = 1 + y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{8}{2}$

$x = 1 + y$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{8}{2}$

$x = 1 + y$

$y = \frac{8}{2}$

$x = 1 + y$

$y = \frac{8}{2}$

$x = 1 + y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $8$ by $2$ .

$y = 4$

$x = 1 + y$

$y = 4$

$x = 1 + y$

$y = 4$

$x = 1 + y$

$y = 4$

$x = 1 + y$

Step 4

Replace all occurrences of $y$ with $4$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 1 + y$ with $4$ .

$x = 1 + 4$

$y = 4$

Step 4.2

Simplify $x = 1 + 4$ .

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Remove parentheses.

$x = 1 + 4$

$y = 4$

$x = 1 + 4$

$y = 4$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Add $1$ and $4$ .

$x = 5$

$y = 4$

$x = 5$

$y = 4$

$x = 5$

$y = 4$

$x = 5$

$y = 4$

Step 5

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

$(5, 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(5, 4)$

Equation Form:

$x = 5, y = 4$

Step 7