Solve the System of Equations y=-3x^2-19x+9 x+y=9

$y = - 3 x^{2} - 19 x + 9$ $x + y = 9$

Replace all occurrences of $y$ with $- 3 x^{2} - 19 x + 9$ in each equation.

Replace all occurrences of $y$ in $x + y = 9$ with $- 3 x^{2} - 19 x + 9$ .

$x - 3 x^{2} - 19 x + 9 = 9$

$y = - 3 x^{2} - 19 x + 9$

Simplify the left side.

Simplify $x - 3 x^{2} - 19 x + 9$ .

Remove parentheses.

$x - 3 x^{2} - 19 x + 9 = 9$

$y = - 3 x^{2} - 19 x + 9$

Subtract $19 x$ from $x$ .

$- 3 x^{2} - 18 x + 9 = 9$

$y = - 3 x^{2} - 19 x + 9$

$- 3 x^{2} - 18 x + 9 = 9$

$y = - 3 x^{2} - 19 x + 9$

$- 3 x^{2} - 18 x + 9 = 9$

$y = - 3 x^{2} - 19 x + 9$

$- 3 x^{2} - 18 x + 9 = 9$

$y = - 3 x^{2} - 19 x + 9$

Solve for $x$ in $- 3 x^{2} - 18 x + 9 = 9$ .

Subtract $9$ from both sides of the equation.

$- 3 x^{2} - 18 x + 9 - 9 = 0$

$y = - 3 x^{2} - 19 x + 9$

Combine the opposite terms in $- 3 x^{2} - 18 x + 9 - 9$ .

Subtract $9$ from $9$ .

$- 3 x^{2} - 18 x + 0 = 0$

$y = - 3 x^{2} - 19 x + 9$

Add $- 3 x^{2} - 18 x$ and $0$ .

$- 3 x^{2} - 18 x = 0$

$y = - 3 x^{2} - 19 x + 9$

$- 3 x^{2} - 18 x = 0$

$y = - 3 x^{2} - 19 x + 9$

Factor $- 3 x$ out of $- 3 x^{2} - 18 x$ .

Factor $- 3 x$ out of $- 3 x^{2}$ .

$- 3 x \cdot x - 18 x = 0$

$y = - 3 x^{2} - 19 x + 9$

Factor $- 3 x$ out of $- 18 x$ .

$- 3 x \cdot x - 3 x \cdot 6 = 0$

$y = - 3 x^{2} - 19 x + 9$

Factor $- 3 x$ out of $- 3 x (x) - 3 x (6)$ .

$- 3 x (x + 6) = 0$

$y = - 3 x^{2} - 19 x + 9$

$- 3 x (x + 6) = 0$

$y = - 3 x^{2} - 19 x + 9$

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

$x = 0$

$x + 6 = 0$

$y = - 3 x^{2} - 19 x + 9$

Set $x$ equal to $0$ .

$x = 0$

$y = - 3 x^{2} - 19 x + 9$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

$y = - 3 x^{2} - 19 x + 9$

Subtract $6$ from both sides of the equation.

$x = - 6$

$y = - 3 x^{2} - 19 x + 9$

$x = - 6$

$y = - 3 x^{2} - 19 x + 9$

The final solution is all the values that make $- 3 x (x + 6) = 0$ true.

$x = 0, - 6$

$y = - 3 x^{2} - 19 x + 9$

$x = 0, - 6$

$y = - 3 x^{2} - 19 x + 9$

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

Replace all occurrences of $x$ in $y = - 3 x^{2} - 19 x + 9$ with $0$ .

$y = - 3 {(0)}^{2} - 19 \cdot 0 + 9$

$x = 0$

Simplify the right side.

Simplify $- 3 {(0)}^{2} - 19 \cdot 0 + 9$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$y = - 3 \cdot 0 - 19 \cdot 0 + 9$

$x = 0$

Multiply $- 3$ by $0$ .

$y = 0 - 19 \cdot 0 + 9$

$x = 0$

Multiply $- 19$ by $0$ .

$y = 0 + 0 + 9$

$x = 0$

$y = 0 + 0 + 9$

$x = 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 9$

$x = 0$

Add $0$ and $9$ .

$y = 9$

$x = 0$

$y = 9$

$x = 0$

$y = 9$

$x = 0$

$y = 9$

$x = 0$

$y = 9$

$x = 0$

Replace all occurrences of $x$ with $- 6$ in each equation.

Replace all occurrences of $x$ in $y = - 3 x^{2} - 19 x + 9$ with $- 6$ .

$y = - 3 {(- 6)}^{2} - 19 \cdot - 6 + 9$

$x = - 6$

Simplify the right side.

Simplify $- 3 {(- 6)}^{2} - 19 \cdot - 6 + 9$ .

Simplify each term.

Raise $- 6$ to the power of $2$ .

$y = - 3 \cdot 36 - 19 \cdot - 6 + 9$

$x = - 6$

Multiply $- 3$ by $36$ .

$y = - 108 - 19 \cdot - 6 + 9$

$x = - 6$

Multiply $- 19$ by $- 6$ .

$y = - 108 + 114 + 9$

$x = - 6$

$y = - 108 + 114 + 9$

$x = - 6$

Simplify by adding numbers.

Add $- 108$ and $114$ .

$y = 6 + 9$

$x = - 6$

Add $6$ and $9$ .

$y = 15$

$x = - 6$

$y = 15$

$x = - 6$

$y = 15$

$x = - 6$

$y = 15$

$x = - 6$

$y = 15$

$x = - 6$

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

$(0, 9)$

$(- 6, 15)$

The result can be shown in multiple forms.

Point Form:

$(0, 9), (- 6, 15)$

Equation Form:

$x = 0, y = 9$

$x = - 6, y = 15$

Solve the System of Equations y=-3x^2-19x+9 x+y=9

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