Algebra Examples

$x^{2} + y^{2} = 49$ $y = 7 - x$

Step 1

Reorder $7$ and $- x$ .

$y = - x + 7$

$x^{2} + y^{2} = 49$

Step 2

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

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Step 2.1

Replace all occurrences of $y$ in $x^{2} + y^{2} = 49$ with $- x + 7$ .

$x^{2} + {(- x + 7)}^{2} = 49$

$y = - x + 7$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

$x^{2} + (- x + 7) (- x + 7) = 49$

$y = - x + 7$

Step 2.2.1.1.2

Expand $(- x + 7) (- x + 7)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$x^{2} - x (- x + 7) + 7 (- x + 7) = 49$

$y = - x + 7$

Step 2.2.1.1.2.2

Apply the distributive property.

$x^{2} - x (- x) - x \cdot 7 + 7 (- x + 7) = 49$

$y = - x + 7$

Step 2.2.1.1.2.3

Apply the distributive property.

$x^{2} - x (- x) - x \cdot 7 + 7 (- x) + 7 \cdot 7 = 49$

$y = - x + 7$

$x^{2} - x (- x) - x \cdot 7 + 7 (- x) + 7 \cdot 7 = 49$

$y = - x + 7$

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

Rewrite using the commutative property of multiplication.

$x^{2} - 1 \cdot (- 1 x \cdot x) - x \cdot 7 + 7 (- x) + 7 \cdot 7 = 49$

$y = - x + 7$

Step 2.2.1.1.3.1.2

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

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Step 2.2.1.1.3.1.2.1

Move $x$ .

$x^{2} - 1 \cdot (- 1 (x \cdot x)) - x \cdot 7 + 7 (- x) + 7 \cdot 7 = 49$

$y = - x + 7$

Step 2.2.1.1.3.1.2.2

Multiply $x$ by $x$ .

$x^{2} - 1 \cdot (- 1 x^{2}) - x \cdot 7 + 7 (- x) + 7 \cdot 7 = 49$

$y = - x + 7$

$x^{2} - 1 \cdot (- 1 x^{2}) - x \cdot 7 + 7 (- x) + 7 \cdot 7 = 49$

$y = - x + 7$

Step 2.2.1.1.3.1.3

Multiply $- 1$ by $- 1$ .

$x^{2} + 1 x^{2} - x \cdot 7 + 7 (- x) + 7 \cdot 7 = 49$

$y = - x + 7$

Step 2.2.1.1.3.1.4

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

$x^{2} + x^{2} - x \cdot 7 + 7 (- x) + 7 \cdot 7 = 49$

$y = - x + 7$

Step 2.2.1.1.3.1.5

Multiply $7$ by $- 1$ .

$x^{2} + x^{2} - 7 x + 7 (- x) + 7 \cdot 7 = 49$

$y = - x + 7$

Step 2.2.1.1.3.1.6

Multiply $- 1$ by $7$ .

$x^{2} + x^{2} - 7 x - 7 x + 7 \cdot 7 = 49$

$y = - x + 7$

Step 2.2.1.1.3.1.7

Multiply $7$ by $7$ .

$x^{2} + x^{2} - 7 x - 7 x + 49 = 49$

$y = - x + 7$

$x^{2} + x^{2} - 7 x - 7 x + 49 = 49$

$y = - x + 7$

Step 2.2.1.1.3.2

Subtract $7 x$ from $- 7 x$ .

$x^{2} + x^{2} - 14 x + 49 = 49$

$y = - x + 7$

$x^{2} + x^{2} - 14 x + 49 = 49$

$y = - x + 7$

$x^{2} + x^{2} - 14 x + 49 = 49$

$y = - x + 7$

Step 2.2.1.2

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

$2 x^{2} - 14 x + 49 = 49$

$y = - x + 7$

$2 x^{2} - 14 x + 49 = 49$

$y = - x + 7$

$2 x^{2} - 14 x + 49 = 49$

$y = - x + 7$

$2 x^{2} - 14 x + 49 = 49$

$y = - x + 7$

Step 3

Solve for $x$ in $2 x^{2} - 14 x + 49 = 49$ .

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Step 3.1

Subtract $49$ from both sides of the equation.

$2 x^{2} - 14 x + 49 - 49 = 0$

$y = - x + 7$

Step 3.2

Combine the opposite terms in $2 x^{2} - 14 x + 49 - 49$ .

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Step 3.2.1

Subtract $49$ from $49$ .

$2 x^{2} - 14 x + 0 = 0$

$y = - x + 7$

Step 3.2.2

Add $2 x^{2} - 14 x$ and $0$ .

$2 x^{2} - 14 x = 0$

$y = - x + 7$

$2 x^{2} - 14 x = 0$

$y = - x + 7$

Step 3.3

Factor $2 x$ out of $2 x^{2} - 14 x$ .

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Step 3.3.1

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

$2 x (x) - 14 x = 0$

$y = - x + 7$

Step 3.3.2

Factor $2 x$ out of $- 14 x$ .

$2 x (x) + 2 x (- 7) = 0$

$y = - x + 7$

Step 3.3.3

Factor $2 x$ out of $2 x (x) + 2 x (- 7)$ .

$2 x (x - 7) = 0$

$y = - x + 7$

$2 x (x - 7) = 0$

$y = - x + 7$

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 $0$ .

$x = 0$

$x - 7 = 0$

$y = - x + 7$

Step 3.5

Set $x$ equal to $0$ .

$x = 0$

$y = - x + 7$

Step 3.6

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

$y = - x + 7$

Step 3.6.2

Add $7$ to both sides of the equation.

$x = 7$

$y = - x + 7$

$x = 7$

$y = - x + 7$

Step 3.7

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

$x = 0, 7$

$y = - x + 7$

$x = 0, 7$

$y = - x + 7$

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 $y = - x + 7$ with $0$ .

$y = - (0) + 7$

$x = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- (0) + 7$ .

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Step 4.2.1.1

Multiply $- 1$ by $0$ .

$y = 0 + 7$

$x = 0$

Step 4.2.1.2

Add $0$ and $7$ .

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

$y = 7$

$x = 0$

Step 5

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

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Step 5.1

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

$y = - (7) + 7$

$x = 7$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- (7) + 7$ .

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Step 5.2.1.1

Multiply $- 1$ by $7$ .

$y = - 7 + 7$

$x = 7$

Step 5.2.1.2

Add $- 7$ and $7$ .

$y = 0$

$x = 7$

$y = 0$

$x = 7$

$y = 0$

$x = 7$

$y = 0$

$x = 7$

Step 6

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

$(0, 7)$

$(7, 0)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(0, 7), (7, 0)$

Equation Form:

$x = 0, y = 7$

$x = 7, y = 0$

Step 8