Algebra Examples

${(x + 3)}^{2} + {(y - 5)}^{2} = 65$ $- 2 x + y = - 4$

Step 1

Add $2 x$ to both sides of the equation.

$y = - 4 + 2 x$

${(x + 3)}^{2} + {(y - 5)}^{2} = 65$

Step 2

Replace all occurrences of $y$ with $- 4 + 2 x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in ${(x + 3)}^{2} + {(y - 5)}^{2} = 65$ with $- 4 + 2 x$ .

${(x + 3)}^{2} + {((- 4 + 2 x) - 5)}^{2} = 65$

$y = - 4 + 2 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(x + 3)}^{2} + {((- 4 + 2 x) - 5)}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

$(x + 3) (x + 3) + {((- 4 + 2 x) - 5)}^{2} = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.2

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

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Step 2.2.1.1.2.1

Apply the distributive property.

$x (x + 3) + 3 (x + 3) + {((- 4 + 2 x) - 5)}^{2} = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot 3 + 3 (x + 3) + {((- 4 + 2 x) - 5)}^{2} = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 + {((- 4 + 2 x) - 5)}^{2} = 65$

$y = - 4 + 2 x$

$x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 + {((- 4 + 2 x) - 5)}^{2} = 65$

$y = - 4 + 2 x$

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 $x$ by $x$ .

$x^{2} + x \cdot 3 + 3 x + 3 \cdot 3 + {((- 4 + 2 x) - 5)}^{2} = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.3.1.2

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

$x^{2} + 3 \cdot x + 3 x + 3 \cdot 3 + {((- 4 + 2 x) - 5)}^{2} = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.3.1.3

Multiply $3$ by $3$ .

$x^{2} + 3 x + 3 x + 9 + {((- 4 + 2 x) - 5)}^{2} = 65$

$y = - 4 + 2 x$

$x^{2} + 3 x + 3 x + 9 + {((- 4 + 2 x) - 5)}^{2} = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.3.2

Add $3 x$ and $3 x$ .

$x^{2} + 6 x + 9 + {((- 4 + 2 x) - 5)}^{2} = 65$

$y = - 4 + 2 x$

$x^{2} + 6 x + 9 + {((- 4 + 2 x) - 5)}^{2} = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.4

Subtract $5$ from $- 4$ .

$x^{2} + 6 x + 9 + {(2 x - 9)}^{2} = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.5

Rewrite ${(2 x - 9)}^{2}$ as $(2 x - 9) (2 x - 9)$ .

$x^{2} + 6 x + 9 + (2 x - 9) (2 x - 9) = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.6

Expand $(2 x - 9) (2 x - 9)$ using the FOIL Method.

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Step 2.2.1.1.6.1

Apply the distributive property.

$x^{2} + 6 x + 9 + 2 x (2 x - 9) - 9 (2 x - 9) = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.6.2

Apply the distributive property.

$x^{2} + 6 x + 9 + 2 x (2 x) + 2 x \cdot - 9 - 9 (2 x - 9) = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.6.3

Apply the distributive property.

$x^{2} + 6 x + 9 + 2 x (2 x) + 2 x \cdot - 9 - 9 (2 x) - 9 \cdot - 9 = 65$

$y = - 4 + 2 x$

$x^{2} + 6 x + 9 + 2 x (2 x) + 2 x \cdot - 9 - 9 (2 x) - 9 \cdot - 9 = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.7

Simplify and combine like terms.

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Step 2.2.1.1.7.1

Simplify each term.

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Step 2.2.1.1.7.1.1

Rewrite using the commutative property of multiplication.

$x^{2} + 6 x + 9 + 2 \cdot (2 x \cdot x) + 2 x \cdot - 9 - 9 (2 x) - 9 \cdot - 9 = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.7.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.1.1.7.1.2.1

Move $x$ .

$x^{2} + 6 x + 9 + 2 \cdot (2 (x \cdot x)) + 2 x \cdot - 9 - 9 (2 x) - 9 \cdot - 9 = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.7.1.2.2

Multiply $x$ by $x$ .

$x^{2} + 6 x + 9 + 2 \cdot (2 x^{2}) + 2 x \cdot - 9 - 9 (2 x) - 9 \cdot - 9 = 65$

$y = - 4 + 2 x$

$x^{2} + 6 x + 9 + 2 \cdot (2 x^{2}) + 2 x \cdot - 9 - 9 (2 x) - 9 \cdot - 9 = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.7.1.3

Multiply $2$ by $2$ .

$x^{2} + 6 x + 9 + 4 x^{2} + 2 x \cdot - 9 - 9 (2 x) - 9 \cdot - 9 = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.7.1.4

Multiply $- 9$ by $2$ .

$x^{2} + 6 x + 9 + 4 x^{2} - 18 x - 9 (2 x) - 9 \cdot - 9 = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.7.1.5

Multiply $2$ by $- 9$ .

$x^{2} + 6 x + 9 + 4 x^{2} - 18 x - 18 x - 9 \cdot - 9 = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.7.1.6

Multiply $- 9$ by $- 9$ .

$x^{2} + 6 x + 9 + 4 x^{2} - 18 x - 18 x + 81 = 65$

$y = - 4 + 2 x$

$x^{2} + 6 x + 9 + 4 x^{2} - 18 x - 18 x + 81 = 65$

$y = - 4 + 2 x$

Step 2.2.1.1.7.2

Subtract $18 x$ from $- 18 x$ .

$x^{2} + 6 x + 9 + 4 x^{2} - 36 x + 81 = 65$

$y = - 4 + 2 x$

$x^{2} + 6 x + 9 + 4 x^{2} - 36 x + 81 = 65$

$y = - 4 + 2 x$

$x^{2} + 6 x + 9 + 4 x^{2} - 36 x + 81 = 65$

$y = - 4 + 2 x$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $x^{2}$ and $4 x^{2}$ .

$5 x^{2} + 6 x + 9 - 36 x + 81 = 65$

$y = - 4 + 2 x$

Step 2.2.1.2.2

Subtract $36 x$ from $6 x$ .

$5 x^{2} - 30 x + 9 + 81 = 65$

$y = - 4 + 2 x$

Step 2.2.1.2.3

Add $9$ and $81$ .

$5 x^{2} - 30 x + 90 = 65$

$y = - 4 + 2 x$

$5 x^{2} - 30 x + 90 = 65$

$y = - 4 + 2 x$

$5 x^{2} - 30 x + 90 = 65$

$y = - 4 + 2 x$

$5 x^{2} - 30 x + 90 = 65$

$y = - 4 + 2 x$

$5 x^{2} - 30 x + 90 = 65$

$y = - 4 + 2 x$

Step 3

Solve for $x$ in $5 x^{2} - 30 x + 90 = 65$ .

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Step 3.1

Subtract $65$ from both sides of the equation.

$5 x^{2} - 30 x + 90 - 65 = 0$

$y = - 4 + 2 x$

Step 3.2

Subtract $65$ from $90$ .

$5 x^{2} - 30 x + 25 = 0$

$y = - 4 + 2 x$

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor $5$ out of $5 x^{2} - 30 x + 25$ .

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Step 3.3.1.1

Factor $5$ out of $5 x^{2}$ .

$5 (x^{2}) - 30 x + 25 = 0$

$y = - 4 + 2 x$

Step 3.3.1.2

Factor $5$ out of $- 30 x$ .

$5 (x^{2}) + 5 (- 6 x) + 25 = 0$

$y = - 4 + 2 x$

Step 3.3.1.3

Factor $5$ out of $25$ .

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

$y = - 4 + 2 x$

Step 3.3.1.4

Factor $5$ out of $5 x^{2} + 5 (- 6 x)$ .

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

$y = - 4 + 2 x$

Step 3.3.1.5

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

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

$y = - 4 + 2 x$

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

$y = - 4 + 2 x$

Step 3.3.2

Factor.

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Step 3.3.2.1

Factor $x^{2} - 6 x + 5$ 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 $- 6$ .

$- 5, - 1$

$y = - 4 + 2 x$

Step 3.3.2.1.2

Write the factored form using these integers.

$5 ((x - 5) (x - 1)) = 0$

$y = - 4 + 2 x$

$5 ((x - 5) (x - 1)) = 0$

$y = - 4 + 2 x$

Step 3.3.2.2

Remove unnecessary parentheses.

$5 (x - 5) (x - 1) = 0$

$y = - 4 + 2 x$

$5 (x - 5) (x - 1) = 0$

$y = - 4 + 2 x$

$5 (x - 5) (x - 1) = 0$

$y = - 4 + 2 x$

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 $0$ .

$x - 5 = 0$

$x - 1 = 0$

$y = - 4 + 2 x$

Step 3.5

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

$y = - 4 + 2 x$

Step 3.5.2

Add $5$ to both sides of the equation.

$x = 5$

$y = - 4 + 2 x$

$x = 5$

$y = - 4 + 2 x$

Step 3.6

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

$y = - 4 + 2 x$

Step 3.6.2

Add $1$ to both sides of the equation.

$x = 1$

$y = - 4 + 2 x$

$x = 1$

$y = - 4 + 2 x$

Step 3.7

The final solution is all the values that make $5 (x - 5) (x - 1) = 0$ true.

$x = 5, 1$

$y = - 4 + 2 x$

$x = 5, 1$

$y = - 4 + 2 x$

Step 4

Replace all occurrences of $x$ with $5$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - 4 + 2 x$ with $5$ .

$y = - 4 + 2 (5)$

$x = 5$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 4 + 2 (5)$ .

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Step 4.2.1.1

Multiply $2$ by $5$ .

$y = - 4 + 10$

$x = 5$

Step 4.2.1.2

Add $- 4$ and $10$ .

$y = 6$

$x = 5$

$y = 6$

$x = 5$

$y = 6$

$x = 5$

$y = 6$

$x = 5$

Step 5

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

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Step 5.1

Replace all occurrences of $x$ in $y = - 4 + 2 x$ with $1$ .

$y = - 4 + 2 (1)$

$x = 1$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 4 + 2 (1)$ .

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Step 5.2.1.1

Multiply $2$ by $1$ .

$y = - 4 + 2$

$x = 1$

Step 5.2.1.2

Add $- 4$ and $2$ .

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

Step 6

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

$(5, 6)$

$(1, - 2)$

Step 7

The result can be shown in multiple forms.

Point Form:

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

Equation Form:

$x = 5, y = 6$

$x = 1, y = - 2$

Step 8