Algebra Examples

$y = - x^{2} - 13 x - 3$ $2 x + y = 15$

Step 1

Replace all occurrences of $y$ with $- x^{2} - 13 x - 3$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $2 x + y = 15$ with $- x^{2} - 13 x - 3$ .

$2 x - x^{2} - 13 x - 3 = 15$

$y = - x^{2} - 13 x - 3$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $2 x - x^{2} - 13 x - 3$ .

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Step 1.2.1.1

Remove parentheses.

$2 x - x^{2} - 13 x - 3 = 15$

$y = - x^{2} - 13 x - 3$

Step 1.2.1.2

Subtract $13 x$ from $2 x$ .

$- x^{2} - 11 x - 3 = 15$

$y = - x^{2} - 13 x - 3$

$- x^{2} - 11 x - 3 = 15$

$y = - x^{2} - 13 x - 3$

$- x^{2} - 11 x - 3 = 15$

$y = - x^{2} - 13 x - 3$

$- x^{2} - 11 x - 3 = 15$

$y = - x^{2} - 13 x - 3$

Step 2

Solve for $x$ in $- x^{2} - 11 x - 3 = 15$ .

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Step 2.1

Subtract $15$ from both sides of the equation.

$- x^{2} - 11 x - 3 - 15 = 0$

$y = - x^{2} - 13 x - 3$

Step 2.2

Subtract $15$ from $- 3$ .

$- x^{2} - 11 x - 18 = 0$

$y = - x^{2} - 13 x - 3$

Step 2.3

Factor the left side of the equation.

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Step 2.3.1

Factor $- 1$ out of $- x^{2} - 11 x - 18$ .

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Step 2.3.1.1

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) - 11 x - 18 = 0$

$y = - x^{2} - 13 x - 3$

Step 2.3.1.2

Factor $- 1$ out of $- 11 x$ .

$- (x^{2}) - (11 x) - 18 = 0$

$y = - x^{2} - 13 x - 3$

Step 2.3.1.3

Rewrite $- 18$ as $- 1 (18)$ .

$- (x^{2}) - (11 x) - 1 \cdot 18 = 0$

$y = - x^{2} - 13 x - 3$

Step 2.3.1.4

Factor $- 1$ out of $- (x^{2}) - (11 x)$ .

$- (x^{2} + 11 x) - 1 \cdot 18 = 0$

$y = - x^{2} - 13 x - 3$

Step 2.3.1.5

Factor $- 1$ out of $- (x^{2} + 11 x) - 1 (18)$ .

$- (x^{2} + 11 x + 18) = 0$

$y = - x^{2} - 13 x - 3$

$- (x^{2} + 11 x + 18) = 0$

$y = - x^{2} - 13 x - 3$

Step 2.3.2

Factor.

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Step 2.3.2.1

Factor $x^{2} + 11 x + 18$ using the AC method.

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Step 2.3.2.1.1

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 $18$ and whose sum is $11$ .

$2, 9$

$y = - x^{2} - 13 x - 3$

Step 2.3.2.1.2

Write the factored form using these integers.

$- ((x + 2) (x + 9)) = 0$

$y = - x^{2} - 13 x - 3$

$- ((x + 2) (x + 9)) = 0$

$y = - x^{2} - 13 x - 3$

Step 2.3.2.2

Remove unnecessary parentheses.

$- (x + 2) (x + 9) = 0$

$y = - x^{2} - 13 x - 3$

$- (x + 2) (x + 9) = 0$

$y = - x^{2} - 13 x - 3$

$- (x + 2) (x + 9) = 0$

$y = - x^{2} - 13 x - 3$

Step 2.4

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

$x + 2 = 0$

$x + 9 = 0$

$y = - x^{2} - 13 x - 3$

Step 2.5

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

$y = - x^{2} - 13 x - 3$

Step 2.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

$y = - x^{2} - 13 x - 3$

$x = - 2$

$y = - x^{2} - 13 x - 3$

Step 2.6

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

$y = - x^{2} - 13 x - 3$

Step 2.6.2

Subtract $9$ from both sides of the equation.

$x = - 9$

$y = - x^{2} - 13 x - 3$

$x = - 9$

$y = - x^{2} - 13 x - 3$

Step 2.7

The final solution is all the values that make $- (x + 2) (x + 9) = 0$ true.

$x = - 2, - 9$

$y = - x^{2} - 13 x - 3$

$x = - 2, - 9$

$y = - x^{2} - 13 x - 3$

Step 3

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

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Step 3.1

Replace all occurrences of $x$ in $y = - x^{2} - 13 x - 3$ with $- 2$ .

$y = - {(- 2)}^{2} - 13 \cdot - 2 - 3$

$x = - 2$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- {(- 2)}^{2} - 13 \cdot - 2 - 3$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

Raise $- 2$ to the power of $2$ .

$y = - 1 \cdot 4 - 13 \cdot - 2 - 3$

$x = - 2$

Step 3.2.1.1.2

Multiply $- 1$ by $4$ .

$y = - 4 - 13 \cdot - 2 - 3$

$x = - 2$

Step 3.2.1.1.3

Multiply $- 13$ by $- 2$ .

$y = - 4 + 26 - 3$

$x = - 2$

$y = - 4 + 26 - 3$

$x = - 2$

Step 3.2.1.2

Simplify by adding and subtracting.

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Step 3.2.1.2.1

Add $- 4$ and $26$ .

$y = 22 - 3$

$x = - 2$

Step 3.2.1.2.2

Subtract $3$ from $22$ .

$y = 19$

$x = - 2$

$y = 19$

$x = - 2$

$y = 19$

$x = - 2$

$y = 19$

$x = - 2$

$y = 19$

$x = - 2$

Step 4

Replace all occurrences of $x$ with $- 9$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - x^{2} - 13 x - 3$ with $- 9$ .

$y = - {(- 9)}^{2} - 13 \cdot - 9 - 3$

$x = - 9$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- {(- 9)}^{2} - 13 \cdot - 9 - 3$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Raise $- 9$ to the power of $2$ .

$y = - 1 \cdot 81 - 13 \cdot - 9 - 3$

$x = - 9$

Step 4.2.1.1.2

Multiply $- 1$ by $81$ .

$y = - 81 - 13 \cdot - 9 - 3$

$x = - 9$

Step 4.2.1.1.3

Multiply $- 13$ by $- 9$ .

$y = - 81 + 117 - 3$

$x = - 9$

$y = - 81 + 117 - 3$

$x = - 9$

Step 4.2.1.2

Simplify by adding and subtracting.

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Step 4.2.1.2.1

Add $- 81$ and $117$ .

$y = 36 - 3$

$x = - 9$

Step 4.2.1.2.2

Subtract $3$ from $36$ .

$y = 33$

$x = - 9$

$y = 33$

$x = - 9$

$y = 33$

$x = - 9$

$y = 33$

$x = - 9$

$y = 33$

$x = - 9$

Step 5

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

$(- 2, 19)$

$(- 9, 33)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 2, 19), (- 9, 33)$

Equation Form:

$x = - 2, y = 19$

$x = - 9, y = 33$

Step 7