Solve the System of Equations y=-x^2+1 and y=x^2-1

$y = - x^{2} + 1$ and $y = x^{2} - 1$

Eliminate the equal sides of each equation and combine.

$- x^{2} + 1 = x^{2} - 1$

Solve $- x^{2} + 1 = x^{2} - 1$ for $x$ .

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

Subtract $x^{2}$ from both sides of the equation.

$- x^{2} + 1 - x^{2} = - 1$

Subtract $x^{2}$ from $- x^{2}$ .

$- 2 x^{2} + 1 = - 1$

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

Subtract $1$ from both sides of the equation.

$- 2 x^{2} = - 1 - 1$

Subtract $1$ from $- 1$ .

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

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

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Divide $- 2$ by $- 2$ .

$x^{2} = 1$

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

$x = \pm \sqrt{1}$

Any root of $1$ is $1$ .

$x = \pm 1$

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.

$x = 1$

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

$x = - 1$

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

$x = 1, - 1$

Evaluate $y$ when $x = 1$ .

Substitute $1$ for $x$ .

$y = {(1)}^{2} - 1$

Substitute $1$ for $x$ in $y = {(1)}^{2} - 1$ and solve for $y$ .

Remove parentheses.

$y = 1^{2} - 1$

Simplify $1^{2} - 1$ .

One to any power is one.

$y = 1 - 1$

Subtract $1$ from $1$ .

$y = 0$

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

$(1, 0)$

$(- 1, 0)$

The result can be shown in multiple forms.

Point Form:

$(1, 0), (- 1, 0)$

Equation Form:

$x = 1, y = 0$

$x = - 1, y = 0$

Solve the System of Equations y=-x^2+1 and y=x^2-1

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