Algebra Examples

${(x + 1)}^{2} + y^{2} = 26$ $x + y = - 7$

Step 1

Subtract $y$ from both sides of the equation.

$x = - 7 - y$

${(x + 1)}^{2} + y^{2} = 26$

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 ${(x + 1)}^{2} + y^{2} = 26$ with $- 7 - y$ .

${((- 7 - y) + 1)}^{2} + y^{2} = 26$

$x = - 7 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${((- 7 - y) + 1)}^{2} + y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Add $- 7$ and $1$ .

${(- y - 6)}^{2} + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.2

Rewrite ${(- y - 6)}^{2}$ as $(- y - 6) (- y - 6)$ .

$(- y - 6) (- y - 6) + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.3

Expand $(- y - 6) (- y - 6)$ using the FOIL Method.

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Step 2.2.1.1.3.1

Apply the distributive property.

$- y (- y - 6) - 6 (- y - 6) + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.3.2

Apply the distributive property.

$- y (- y) - y \cdot - 6 - 6 (- y - 6) + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.3.3

Apply the distributive property.

$- y (- y) - y \cdot - 6 - 6 (- y) - 6 \cdot - 6 + y^{2} = 26$

$x = - 7 - y$

$- y (- y) - y \cdot - 6 - 6 (- y) - 6 \cdot - 6 + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.4

Simplify and combine like terms.

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Step 2.2.1.1.4.1

Simplify each term.

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Step 2.2.1.1.4.1.1

Rewrite using the commutative property of multiplication.

$- 1 \cdot (- 1 y \cdot y) - y \cdot - 6 - 6 (- y) - 6 \cdot - 6 + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.4.1.2

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

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Step 2.2.1.1.4.1.2.1

Move $y$ .

$- 1 \cdot (- 1 (y \cdot y)) - y \cdot - 6 - 6 (- y) - 6 \cdot - 6 + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.4.1.2.2

Multiply $y$ by $y$ .

$- 1 \cdot (- 1 y^{2}) - y \cdot - 6 - 6 (- y) - 6 \cdot - 6 + y^{2} = 26$

$x = - 7 - y$

$- 1 \cdot (- 1 y^{2}) - y \cdot - 6 - 6 (- y) - 6 \cdot - 6 + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.4.1.3

Multiply $- 1$ by $- 1$ .

$1 y^{2} - y \cdot - 6 - 6 (- y) - 6 \cdot - 6 + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.4.1.4

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

$y^{2} - y \cdot - 6 - 6 (- y) - 6 \cdot - 6 + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.4.1.5

Multiply $- 6$ by $- 1$ .

$y^{2} + 6 y - 6 (- y) - 6 \cdot - 6 + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.4.1.6

Multiply $- 1$ by $- 6$ .

$y^{2} + 6 y + 6 y - 6 \cdot - 6 + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.4.1.7

Multiply $- 6$ by $- 6$ .

$y^{2} + 6 y + 6 y + 36 + y^{2} = 26$

$x = - 7 - y$

$y^{2} + 6 y + 6 y + 36 + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.1.4.2

Add $6 y$ and $6 y$ .

$y^{2} + 12 y + 36 + y^{2} = 26$

$x = - 7 - y$

$y^{2} + 12 y + 36 + y^{2} = 26$

$x = - 7 - y$

$y^{2} + 12 y + 36 + y^{2} = 26$

$x = - 7 - y$

Step 2.2.1.2

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

$2 y^{2} + 12 y + 36 = 26$

$x = - 7 - y$

$2 y^{2} + 12 y + 36 = 26$

$x = - 7 - y$

$2 y^{2} + 12 y + 36 = 26$

$x = - 7 - y$

$2 y^{2} + 12 y + 36 = 26$

$x = - 7 - y$

Step 3

Solve for $y$ in $2 y^{2} + 12 y + 36 = 26$ .

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Step 3.1

Subtract $26$ from both sides of the equation.

$2 y^{2} + 12 y + 36 - 26 = 0$

$x = - 7 - y$

Step 3.2

Subtract $26$ from $36$ .

$2 y^{2} + 12 y + 10 = 0$

$x = - 7 - y$

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor $2$ out of $2 y^{2} + 12 y + 10$ .

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Step 3.3.1.1

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

$2 (y^{2}) + 12 y + 10 = 0$

$x = - 7 - y$

Step 3.3.1.2

Factor $2$ out of $12 y$ .

$2 (y^{2}) + 2 (6 y) + 10 = 0$

$x = - 7 - y$

Step 3.3.1.3

Factor $2$ out of $10$ .

$2 y^{2} + 2 (6 y) + 2 \cdot 5 = 0$

$x = - 7 - y$

Step 3.3.1.4

Factor $2$ out of $2 y^{2} + 2 (6 y)$ .

$2 (y^{2} + 6 y) + 2 \cdot 5 = 0$

$x = - 7 - y$

Step 3.3.1.5

Factor $2$ out of $2 (y^{2} + 6 y) + 2 \cdot 5$ .

$2 (y^{2} + 6 y + 5) = 0$

$x = - 7 - y$

$2 (y^{2} + 6 y + 5) = 0$

$x = - 7 - y$

Step 3.3.2

Factor.

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Step 3.3.2.1

Factor $y^{2} + 6 y + 5$ using the AC method.

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Step 3.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $5$ and whose sum is $6$ .

$1, 5$

$x = - 7 - y$

Step 3.3.2.1.2

Write the factored form using these integers.

$2 ((y + 1) (y + 5)) = 0$

$x = - 7 - y$

$2 ((y + 1) (y + 5)) = 0$

$x = - 7 - y$

Step 3.3.2.2

Remove unnecessary parentheses.

$2 (y + 1) (y + 5) = 0$

$x = - 7 - y$

$2 (y + 1) (y + 5) = 0$

$x = - 7 - y$

$2 (y + 1) (y + 5) = 0$

$x = - 7 - y$

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 $0$ .

$y + 1 = 0$

$y + 5 = 0$

$x = - 7 - y$

Step 3.5

Set $y + 1$ equal to $0$ and solve for $y$ .

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Step 3.5.1

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

$x = - 7 - y$

Step 3.5.2

Subtract $1$ from both sides of the equation.

$y = - 1$

$x = - 7 - y$

$y = - 1$

$x = - 7 - y$

Step 3.6

Set $y + 5$ equal to $0$ and solve for $y$ .

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Step 3.6.1

Set $y + 5$ equal to $0$ .

$y + 5 = 0$

$x = - 7 - y$

Step 3.6.2

Subtract $5$ from both sides of the equation.

$y = - 5$

$x = - 7 - y$

$y = - 5$

$x = - 7 - y$

Step 3.7

The final solution is all the values that make $2 (y + 1) (y + 5) = 0$ true.

$y = - 1, - 5$

$x = - 7 - y$

$y = - 1, - 5$

$x = - 7 - y$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $x = - 7 - y$ with $- 1$ .

$x = - 7 - (- 1)$

$y = - 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 7 - (- 1)$ .

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Step 4.2.1.1

Multiply $- 1$ by $- 1$ .

$x = - 7 + 1$

$y = - 1$

Step 4.2.1.2

Add $- 7$ and $1$ .

$x = - 6$

$y = - 1$

$x = - 6$

$y = - 1$

$x = - 6$

$y = - 1$

$x = - 6$

$y = - 1$

Step 5

Replace all occurrences of $y$ with $- 5$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = - 7 - y$ with $- 5$ .

$x = - 7 - (- 5)$

$y = - 5$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 7 - (- 5)$ .

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Step 5.2.1.1

Multiply $- 1$ by $- 5$ .

$x = - 7 + 5$

$y = - 5$

Step 5.2.1.2

Add $- 7$ and $5$ .

$x = - 2$

$y = - 5$

$x = - 2$

$y = - 5$

$x = - 2$

$y = - 5$

$x = - 2$

$y = - 5$

Step 6

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

$(- 6, - 1)$

$(- 2, - 5)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- 6, - 1), (- 2, - 5)$

Equation Form:

$x = - 6, y = - 1$

$x = - 2, y = - 5$

Step 8