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

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

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

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

Move all terms containing $x$ to the left side of the equation.

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

$x^{2} + x + 9 + 2 x = 9$

Add $x$ and $2 x$ .

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

Subtract $9$ from both sides of the equation.

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

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

Subtract $9$ from $9$ .

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

Add $x^{2} + 3 x$ and $0$ .

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

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

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

$x \cdot x + 3 x = 0$

Factor $x$ out of $3 x$ .

$x \cdot x + x \cdot 3 = 0$

Factor $x$ out of $x \cdot x + x \cdot 3$ .

$x (x + 3) = 0$

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

$x = 0$

$x + 3 = 0 + y = - 2 x + 9$

Set $x$ equal to $0$ .

$x = 0$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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

$x = 0, - 3$

Step 3

Evaluate $y$ when $x = 0$ .

Substitute $0$ for $x$ .

$y = - 2 \cdot 0 + 9$

Substitute $0$ for $x$ in $y = - 2 \cdot 0 + 9$ and solve for $y$ .

Remove parentheses.

$y = - 2 \cdot 0 + 9$

Simplify $- 2 \cdot 0 + 9$ .

Multiply $- 2$ by $0$ .

$y = 0 + 9$

Add $0$ and $9$ .

$y = 9$

Step 4

Evaluate $y$ when $x = - 3$ .

Substitute $- 3$ for $x$ .

$y = - 2 \cdot - 3 + 9$

Substitute $- 3$ for $x$ in $y = - 2 \cdot - 3 + 9$ and solve for $y$ .

Remove parentheses.

$y = - 2 \cdot - 3 + 9$

Simplify $- 2 \cdot - 3 + 9$ .

Multiply $- 2$ by $- 3$ .

$y = 6 + 9$

Add $6$ and $9$ .

$y = 15$

Step 5

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

$(0, 9)$

$(- 3, 15)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(0, 9), (- 3, 15)$

Equation Form:

$x = 0, y = 9$

$x = - 3, y = 15$

Step 7

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

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