Solve the System of Equations y=x^2+3x-7 3x-y=-2

$y = x^{2} + 3 x - 7$ $3 x - y = - 2$

Step 1

Replace all occurrences of $y$ with $x^{2} + 3 x - 7$ in each equation.

Replace all occurrences of $y$ in $3 x - y = - 2$ with $x^{2} + 3 x - 7$ .

$3 x - (x^{2} + 3 x - 7) = - 2$

$y = x^{2} + 3 x - 7$

Simplify the left side.

Simplify $3 x - (x^{2} + 3 x - 7)$ .

Simplify each term.

Apply the distributive property.

$3 x - x^{2} - (3 x) + 7 = - 2$

$y = x^{2} + 3 x - 7$

Simplify.

Multiply $3$ by $- 1$ .

$3 x - x^{2} - 3 x + 7 = - 2$

$y = x^{2} + 3 x - 7$

Multiply $- 1$ by $- 7$ .

$3 x - x^{2} - 3 x + 7 = - 2$

$y = x^{2} + 3 x - 7$

$3 x - x^{2} - 3 x + 7 = - 2$

$y = x^{2} + 3 x - 7$

$3 x - x^{2} - 3 x + 7 = - 2$

$y = x^{2} + 3 x - 7$

Combine the opposite terms in $3 x - x^{2} - 3 x + 7$ .

Subtract $3 x$ from $3 x$ .

$- x^{2} + 0 + 7 = - 2$

$y = x^{2} + 3 x - 7$

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

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

$y = x^{2} + 3 x - 7$

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

$y = x^{2} + 3 x - 7$

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

$y = x^{2} + 3 x - 7$

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

$y = x^{2} + 3 x - 7$

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

$y = x^{2} + 3 x - 7$

Step 2

Solve for $x$ in $- x^{2} + 7 = - 2$ .

Move all terms not containing $x$ to the right side of the equation.

Subtract $7$ from both sides of the equation.

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

$y = x^{2} + 3 x - 7$

Subtract $7$ from $- 2$ .

$- x^{2} = - 9$

$y = x^{2} + 3 x - 7$

$- x^{2} = - 9$

$y = x^{2} + 3 x - 7$

Divide each term in $- x^{2} = - 9$ by $- 1$ and simplify.

Divide each term in $- x^{2} = - 9$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 9}{- 1}$

$y = x^{2} + 3 x - 7$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 9}{- 1}$

$y = x^{2} + 3 x - 7$

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

$x^{2} = \frac{- 9}{- 1}$

$y = x^{2} + 3 x - 7$

$x^{2} = \frac{- 9}{- 1}$

$y = x^{2} + 3 x - 7$

Simplify the right side.

Divide $- 9$ by $- 1$ .

$x^{2} = 9$

$y = x^{2} + 3 x - 7$

$x^{2} = 9$

$y = x^{2} + 3 x - 7$

$x^{2} = 9$

$y = x^{2} + 3 x - 7$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{9}$

$y = x^{2} + 3 x - 7$

Simplify $\pm \sqrt{9}$ .

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

$y = x^{2} + 3 x - 7$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

$y = x^{2} + 3 x - 7$

$x = \pm 3$

$y = x^{2} + 3 x - 7$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

$y = x^{2} + 3 x - 7$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

$y = x^{2} + 3 x - 7$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

$y = x^{2} + 3 x - 7$

$x = 3, - 3$

$y = x^{2} + 3 x - 7$

$x = 3, - 3$

$y = x^{2} + 3 x - 7$

Step 3

Replace all occurrences of $x$ with $3$ in each equation.

Replace all occurrences of $x$ in $y = x^{2} + 3 x - 7$ with $3$ .

$y = {(3)}^{2} + 3 (3) - 7$

$x = 3$

Simplify the right side.

Simplify ${(3)}^{2} + 3 (3) - 7$ .

Simplify each term.

Raise $3$ to the power of $2$ .

$y = 9 + 3 (3) - 7$

$x = 3$

Multiply $3$ by $3$ .

$y = 9 + 9 - 7$

$x = 3$

$y = 9 + 9 - 7$

$x = 3$

Simplify by adding and subtracting.

Add $9$ and $9$ .

$y = 18 - 7$

$x = 3$

Subtract $7$ from $18$ .

$y = 11$

$x = 3$

$y = 11$

$x = 3$

$y = 11$

$x = 3$

$y = 11$

$x = 3$

$y = 11$

$x = 3$

Step 4

Replace all occurrences of $x$ with $- 3$ in each equation.

Replace all occurrences of $x$ in $y = x^{2} + 3 x - 7$ with $- 3$ .

$y = {(- 3)}^{2} + 3 (- 3) - 7$

$x = - 3$

Simplify the right side.

Simplify ${(- 3)}^{2} + 3 (- 3) - 7$ .

Simplify each term.

Raise $- 3$ to the power of $2$ .

$y = 9 + 3 (- 3) - 7$

$x = - 3$

Multiply $3$ by $- 3$ .

$y = 9 - 9 - 7$

$x = - 3$

$y = 9 - 9 - 7$

$x = - 3$

Simplify by subtracting numbers.

Subtract $9$ from $9$ .

$y = 0 - 7$

$x = - 3$

Subtract $7$ from $0$ .

$y = - 7$

$x = - 3$

$y = - 7$

$x = - 3$

$y = - 7$

$x = - 3$

$y = - 7$

$x = - 3$

$y = - 7$

$x = - 3$

Step 5

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

$(3, 11)$

$(- 3, - 7)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(3, 11), (- 3, - 7)$

Equation Form:

$x = 3, y = 11$

$x = - 3, y = - 7$

Step 7

Solve the System of Equations y=x^2+3x-7 3x-y=-2

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