Algebra Examples

$y = x^{2} - 1$ $- 7 = - x^{2} - y$

Step 1

Replace all occurrences of $y$ with $x^{2} - 1$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $- 7 = - x^{2} - y$ with $x^{2} - 1$ .

$- 7 = - x^{2} - (x^{2} - 1)$

$y = x^{2} - 1$

Step 1.2

Simplify the right side.

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Step 1.2.1

Simplify $- x^{2} - (x^{2} - 1)$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Apply the distributive property.

$- 7 = - x^{2} - x^{2} + 1$

$y = x^{2} - 1$

Step 1.2.1.1.2

Multiply $- 1$ by $- 1$ .

$- 7 = - x^{2} - x^{2} + 1$

$y = x^{2} - 1$

$- 7 = - x^{2} - x^{2} + 1$

$y = x^{2} - 1$

Step 1.2.1.2

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

$- 7 = - 2 x^{2} + 1$

$y = x^{2} - 1$

$- 7 = - 2 x^{2} + 1$

$y = x^{2} - 1$

$- 7 = - 2 x^{2} + 1$

$y = x^{2} - 1$

$- 7 = - 2 x^{2} + 1$

$y = x^{2} - 1$

Step 2

Solve for $x$ in $- 7 = - 2 x^{2} + 1$ .

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Step 2.1

Rewrite the equation as $- 2 x^{2} + 1 = - 7$ .

$- 2 x^{2} + 1 = - 7$

$y = x^{2} - 1$

Step 2.2

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

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Step 2.2.1

Subtract $1$ from both sides of the equation.

$- 2 x^{2} = - 7 - 1$

$y = x^{2} - 1$

Step 2.2.2

Subtract $1$ from $- 7$ .

$- 2 x^{2} = - 8$

$y = x^{2} - 1$

$- 2 x^{2} = - 8$

$y = x^{2} - 1$

Step 2.3

Divide each term in $- 2 x^{2} = - 8$ by $- 2$ and simplify.

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Step 2.3.1

Divide each term in $- 2 x^{2} = - 8$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{- 8}{- 2}$

$y = x^{2} - 1$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{- 8}{- 2}$

$y = x^{2} - 1$

Step 2.3.2.1.2

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

$x^{2} = \frac{- 8}{- 2}$

$y = x^{2} - 1$

$x^{2} = \frac{- 8}{- 2}$

$y = x^{2} - 1$

$x^{2} = \frac{- 8}{- 2}$

$y = x^{2} - 1$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 8$ by $- 2$ .

$x^{2} = 4$

$y = x^{2} - 1$

$x^{2} = 4$

$y = x^{2} - 1$

$x^{2} = 4$

$y = x^{2} - 1$

Step 2.4

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

$x = \pm \sqrt{4}$

$y = x^{2} - 1$

Step 2.5

Simplify $\pm \sqrt{4}$ .

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Step 2.5.1

Rewrite $4$ as $2^{2}$ .

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

$y = x^{2} - 1$

Step 2.5.2

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

$x = \pm 2$

$y = x^{2} - 1$

$x = \pm 2$

$y = x^{2} - 1$

Step 2.6

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

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Step 2.6.1

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

$x = 2$

$y = x^{2} - 1$

Step 2.6.2

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

$x = - 2$

$y = x^{2} - 1$

Step 2.6.3

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

$x = 2, - 2$

$y = x^{2} - 1$

$x = 2, - 2$

$y = x^{2} - 1$

$x = 2, - 2$

$y = x^{2} - 1$

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 $y = x^{2} - 1$ with $2$ .

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

$x = 2$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify ${(2)}^{2} - 1$ .

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Step 3.2.1.1

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

$y = 4 - 1$

$x = 2$

Step 3.2.1.2

Subtract $1$ from $4$ .

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

Step 4

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

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Step 4.1

Replace all occurrences of $x$ in $y = x^{2} - 1$ with $- 2$ .

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

$x = - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify ${(- 2)}^{2} - 1$ .

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Step 4.2.1.1

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

$y = 4 - 1$

$x = - 2$

Step 4.2.1.2

Subtract $1$ from $4$ .

$y = 3$

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

$x = - 2$

Step 5

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

$(2, 3)$

$(- 2, 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, 3), (- 2, 3)$

Equation Form:

$x = 2, y = 3$

$x = - 2, y = 3$

Step 7