Algebra Examples

$x^{2} + {(y + 3)}^{2} = 25$ $x - y = 4$

Step 1

Add $y$ to both sides of the equation.

$x = 4 + y$

$x^{2} + {(y + 3)}^{2} = 25$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $x^{2} + {(y + 3)}^{2} = 25$ with $4 + y$ .

${(4 + y)}^{2} + {(y + 3)}^{2} = 25$

$x = 4 + y$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(4 + y)}^{2}$ as $(4 + y) (4 + y)$ .

$(4 + y) (4 + y) + {(y + 3)}^{2} = 25$

$x = 4 + y$

Step 2.2.1.1.2

Expand $(4 + y) (4 + y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$4 (4 + y) + y (4 + y) + {(y + 3)}^{2} = 25$

$x = 4 + y$

Step 2.2.1.1.2.2

Apply the distributive property.

$4 \cdot 4 + 4 y + y (4 + y) + {(y + 3)}^{2} = 25$

$x = 4 + y$

Step 2.2.1.1.2.3

Apply the distributive property.

$4 \cdot 4 + 4 y + y \cdot 4 + y \cdot y + {(y + 3)}^{2} = 25$

$x = 4 + y$

$4 \cdot 4 + 4 y + y \cdot 4 + y \cdot y + {(y + 3)}^{2} = 25$

$x = 4 + y$

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 $4$ by $4$ .

$16 + 4 y + y \cdot 4 + y \cdot y + {(y + 3)}^{2} = 25$

$x = 4 + y$

Step 2.2.1.1.3.1.2

Move $4$ to the left of $y$ .

$16 + 4 y + 4 \cdot y + y \cdot y + {(y + 3)}^{2} = 25$

$x = 4 + y$

Step 2.2.1.1.3.1.3

Multiply $y$ by $y$ .

$16 + 4 y + 4 y + y^{2} + {(y + 3)}^{2} = 25$

$x = 4 + y$

$16 + 4 y + 4 y + y^{2} + {(y + 3)}^{2} = 25$

$x = 4 + y$

Step 2.2.1.1.3.2

Add $4 y$ and $4 y$ .

$16 + 8 y + y^{2} + {(y + 3)}^{2} = 25$

$x = 4 + y$

$16 + 8 y + y^{2} + {(y + 3)}^{2} = 25$

$x = 4 + y$

Step 2.2.1.1.4

Rewrite ${(y + 3)}^{2}$ as $(y + 3) (y + 3)$ .

$16 + 8 y + y^{2} + (y + 3) (y + 3) = 25$

$x = 4 + y$

Step 2.2.1.1.5

Expand $(y + 3) (y + 3)$ using the FOIL Method.

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Step 2.2.1.1.5.1

Apply the distributive property.

$16 + 8 y + y^{2} + y (y + 3) + 3 (y + 3) = 25$

$x = 4 + y$

Step 2.2.1.1.5.2

Apply the distributive property.

$16 + 8 y + y^{2} + y \cdot y + y \cdot 3 + 3 (y + 3) = 25$

$x = 4 + y$

Step 2.2.1.1.5.3

Apply the distributive property.

$16 + 8 y + y^{2} + y \cdot y + y \cdot 3 + 3 y + 3 \cdot 3 = 25$

$x = 4 + y$

$16 + 8 y + y^{2} + y \cdot y + y \cdot 3 + 3 y + 3 \cdot 3 = 25$

$x = 4 + y$

Step 2.2.1.1.6

Simplify and combine like terms.

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Step 2.2.1.1.6.1

Simplify each term.

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Step 2.2.1.1.6.1.1

Multiply $y$ by $y$ .

$16 + 8 y + y^{2} + y^{2} + y \cdot 3 + 3 y + 3 \cdot 3 = 25$

$x = 4 + y$

Step 2.2.1.1.6.1.2

Move $3$ to the left of $y$ .

$16 + 8 y + y^{2} + y^{2} + 3 \cdot y + 3 y + 3 \cdot 3 = 25$

$x = 4 + y$

Step 2.2.1.1.6.1.3

Multiply $3$ by $3$ .

$16 + 8 y + y^{2} + y^{2} + 3 y + 3 y + 9 = 25$

$x = 4 + y$

$16 + 8 y + y^{2} + y^{2} + 3 y + 3 y + 9 = 25$

$x = 4 + y$

Step 2.2.1.1.6.2

Add $3 y$ and $3 y$ .

$16 + 8 y + y^{2} + y^{2} + 6 y + 9 = 25$

$x = 4 + y$

$16 + 8 y + y^{2} + y^{2} + 6 y + 9 = 25$

$x = 4 + y$

$16 + 8 y + y^{2} + y^{2} + 6 y + 9 = 25$

$x = 4 + y$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $16$ and $9$ .

$8 y + y^{2} + y^{2} + 6 y + 25 = 25$

$x = 4 + y$

Step 2.2.1.2.2

Add $8 y$ and $6 y$ .

$y^{2} + y^{2} + 14 y + 25 = 25$

$x = 4 + y$

Step 2.2.1.2.3

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

$2 y^{2} + 14 y + 25 = 25$

$x = 4 + y$

$2 y^{2} + 14 y + 25 = 25$

$x = 4 + y$

$2 y^{2} + 14 y + 25 = 25$

$x = 4 + y$

$2 y^{2} + 14 y + 25 = 25$

$x = 4 + y$

$2 y^{2} + 14 y + 25 = 25$

$x = 4 + y$

Step 3

Solve for $y$ in $2 y^{2} + 14 y + 25 = 25$ .

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Step 3.1

Subtract $25$ from both sides of the equation.

$2 y^{2} + 14 y + 25 - 25 = 0$

$x = 4 + y$

Step 3.2

Combine the opposite terms in $2 y^{2} + 14 y + 25 - 25$ .

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Step 3.2.1

Subtract $25$ from $25$ .

$2 y^{2} + 14 y + 0 = 0$

$x = 4 + y$

Step 3.2.2

Add $2 y^{2} + 14 y$ and $0$ .

$2 y^{2} + 14 y = 0$

$x = 4 + y$

$2 y^{2} + 14 y = 0$

$x = 4 + y$

Step 3.3

Factor $2 y$ out of $2 y^{2} + 14 y$ .

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Step 3.3.1

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

$2 y (y) + 14 y = 0$

$x = 4 + y$

Step 3.3.2

Factor $2 y$ out of $14 y$ .

$2 y (y) + 2 y (7) = 0$

$x = 4 + y$

Step 3.3.3

Factor $2 y$ out of $2 y (y) + 2 y (7)$ .

$2 y (y + 7) = 0$

$x = 4 + y$

$2 y (y + 7) = 0$

$x = 4 + y$

Step 3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y = 0$

$y + 7 = 0$

$x = 4 + y$

Step 3.5

Set $y$ equal to $0$ .

$y = 0$

$x = 4 + y$

Step 3.6

Set $y + 7$ equal to $0$ and solve for $y$ .

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Step 3.6.1

Set $y + 7$ equal to $0$ .

$y + 7 = 0$

$x = 4 + y$

Step 3.6.2

Subtract $7$ from both sides of the equation.

$y = - 7$

$x = 4 + y$

$y = - 7$

$x = 4 + y$

Step 3.7

The final solution is all the values that make $2 y (y + 7) = 0$ true.

$y = 0, - 7$

$x = 4 + y$

$y = 0, - 7$

$x = 4 + y$

Step 4

Replace all occurrences of $y$ with $0$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 4 + y$ with $0$ .

$x = 4 + 0$

$y = 0$

Step 4.2

Simplify $x = 4 + 0$ .

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Remove parentheses.

$x = 4 + 0$

$y = 0$

$x = 4 + 0$

$y = 0$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Add $4$ and $0$ .

$x = 4$

$y = 0$

$x = 4$

$y = 0$

$x = 4$

$y = 0$

$x = 4$

$y = 0$

Step 5

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

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Step 5.1

Replace all occurrences of $y$ in $x = 4 + y$ with $- 7$ .

$x = 4 - 7$

$y = - 7$

Step 5.2

Simplify $x = 4 - 7$ .

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Remove parentheses.

$x = 4 - 7$

$y = - 7$

$x = 4 - 7$

$y = - 7$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Subtract $7$ from $4$ .

$x = - 3$

$y = - 7$

$x = - 3$

$y = - 7$

$x = - 3$

$y = - 7$

$x = - 3$

$y = - 7$

Step 6

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

$(4, 0)$

$(- 3, - 7)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(4, 0), (- 3, - 7)$

Equation Form:

$x = 4, y = 0$

$x = - 3, y = - 7$

Step 8