Algebra Examples

$x^{2} + {(y - 2)}^{2} = 25$ $- 2 x + y = 7$

Step 1

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

$y = 7 + 2 x$

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

Step 2

Replace all occurrences of $y$ with $7 + 2 x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{2} + {(y - 2)}^{2} = 25$ with $7 + 2 x$ .

$x^{2} + {((7 + 2 x) - 2)}^{2} = 25$

$y = 7 + 2 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Subtract $2$ from $7$ .

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

$y = 7 + 2 x$

Step 2.2.1.1.2

Rewrite ${(2 x + 5)}^{2}$ as $(2 x + 5) (2 x + 5)$ .

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

$y = 7 + 2 x$

Step 2.2.1.1.3

Expand $(2 x + 5) (2 x + 5)$ using the FOIL Method.

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Step 2.2.1.1.3.1

Apply the distributive property.

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

$y = 7 + 2 x$

Step 2.2.1.1.3.2

Apply the distributive property.

$x^{2} + 2 x (2 x) + 2 x \cdot 5 + 5 (2 x + 5) = 25$

$y = 7 + 2 x$

Step 2.2.1.1.3.3

Apply the distributive property.

$x^{2} + 2 x (2 x) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5 = 25$

$y = 7 + 2 x$

$x^{2} + 2 x (2 x) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5 = 25$

$y = 7 + 2 x$

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.

$x^{2} + 2 \cdot (2 x \cdot x) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5 = 25$

$y = 7 + 2 x$

Step 2.2.1.1.4.1.2

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

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Step 2.2.1.1.4.1.2.1

Move $x$ .

$x^{2} + 2 \cdot (2 (x \cdot x)) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5 = 25$

$y = 7 + 2 x$

Step 2.2.1.1.4.1.2.2

Multiply $x$ by $x$ .

$x^{2} + 2 \cdot (2 x^{2}) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5 = 25$

$y = 7 + 2 x$

$x^{2} + 2 \cdot (2 x^{2}) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5 = 25$

$y = 7 + 2 x$

Step 2.2.1.1.4.1.3

Multiply $2$ by $2$ .

$x^{2} + 4 x^{2} + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5 = 25$

$y = 7 + 2 x$

Step 2.2.1.1.4.1.4

Multiply $5$ by $2$ .

$x^{2} + 4 x^{2} + 10 x + 5 (2 x) + 5 \cdot 5 = 25$

$y = 7 + 2 x$

Step 2.2.1.1.4.1.5

Multiply $2$ by $5$ .

$x^{2} + 4 x^{2} + 10 x + 10 x + 5 \cdot 5 = 25$

$y = 7 + 2 x$

Step 2.2.1.1.4.1.6

Multiply $5$ by $5$ .

$x^{2} + 4 x^{2} + 10 x + 10 x + 25 = 25$

$y = 7 + 2 x$

$x^{2} + 4 x^{2} + 10 x + 10 x + 25 = 25$

$y = 7 + 2 x$

Step 2.2.1.1.4.2

Add $10 x$ and $10 x$ .

$x^{2} + 4 x^{2} + 20 x + 25 = 25$

$y = 7 + 2 x$

$x^{2} + 4 x^{2} + 20 x + 25 = 25$

$y = 7 + 2 x$

$x^{2} + 4 x^{2} + 20 x + 25 = 25$

$y = 7 + 2 x$

Step 2.2.1.2

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

$5 x^{2} + 20 x + 25 = 25$

$y = 7 + 2 x$

$5 x^{2} + 20 x + 25 = 25$

$y = 7 + 2 x$

$5 x^{2} + 20 x + 25 = 25$

$y = 7 + 2 x$

$5 x^{2} + 20 x + 25 = 25$

$y = 7 + 2 x$

Step 3

Solve for $x$ in $5 x^{2} + 20 x + 25 = 25$ .

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Step 3.1

Subtract $25$ from both sides of the equation.

$5 x^{2} + 20 x + 25 - 25 = 0$

$y = 7 + 2 x$

Step 3.2

Combine the opposite terms in $5 x^{2} + 20 x + 25 - 25$ .

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Step 3.2.1

Subtract $25$ from $25$ .

$5 x^{2} + 20 x + 0 = 0$

$y = 7 + 2 x$

Step 3.2.2

Add $5 x^{2} + 20 x$ and $0$ .

$5 x^{2} + 20 x = 0$

$y = 7 + 2 x$

$5 x^{2} + 20 x = 0$

$y = 7 + 2 x$

Step 3.3

Factor $5 x$ out of $5 x^{2} + 20 x$ .

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Step 3.3.1

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

$5 x (x) + 20 x = 0$

$y = 7 + 2 x$

Step 3.3.2

Factor $5 x$ out of $20 x$ .

$5 x (x) + 5 x (4) = 0$

$y = 7 + 2 x$

Step 3.3.3

Factor $5 x$ out of $5 x (x) + 5 x (4)$ .

$5 x (x + 4) = 0$

$y = 7 + 2 x$

$5 x (x + 4) = 0$

$y = 7 + 2 x$

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 $0$ .

$x = 0$

$x + 4 = 0$

$y = 7 + 2 x$

Step 3.5

Set $x$ equal to $0$ .

$x = 0$

$y = 7 + 2 x$

Step 3.6

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

$y = 7 + 2 x$

Step 3.6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

$y = 7 + 2 x$

$x = - 4$

$y = 7 + 2 x$

Step 3.7

The final solution is all the values that make $5 x (x + 4) = 0$ true.

$x = 0, - 4$

$y = 7 + 2 x$

$x = 0, - 4$

$y = 7 + 2 x$

Step 4

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

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Step 4.1

Replace all occurrences of $x$ in $y = 7 + 2 x$ with $0$ .

$y = 7 + 2 (0)$

$x = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $7 + 2 (0)$ .

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Step 4.2.1.1

Multiply $2$ by $0$ .

$y = 7 + 0$

$x = 0$

Step 4.2.1.2

Add $7$ and $0$ .

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

Step 5

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

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Step 5.1

Replace all occurrences of $x$ in $y = 7 + 2 x$ with $- 4$ .

$y = 7 + 2 (- 4)$

$x = - 4$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $7 + 2 (- 4)$ .

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Step 5.2.1.1

Multiply $2$ by $- 4$ .

$y = 7 - 8$

$x = - 4$

Step 5.2.1.2

Subtract $8$ from $7$ .

$y = - 1$

$x = - 4$

$y = - 1$

$x = - 4$

$y = - 1$

$x = - 4$

$y = - 1$

$x = - 4$

Step 6

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

$(0, 7)$

$(- 4, - 1)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(0, 7), (- 4, - 1)$

Equation Form:

$x = 0, y = 7$

$x = - 4, y = - 1$

Step 8