Algebra Examples

$x = 1 - y^{2}$

Step 1

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

$1 - y^{2} = x$

Step 2

Subtract $1$ from both sides of the equation.

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

Step 3

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

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Step 3.1

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

$\frac{- y^{2}}{- 1} = \frac{x}{- 1} + \frac{- 1}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} = \frac{x}{- 1} + \frac{- 1}{- 1}$

Step 3.2.2

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

$y^{2} = \frac{x}{- 1} + \frac{- 1}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$y^{2} = - 1 \cdot x + \frac{- 1}{- 1}$

Step 3.3.1.2

Rewrite $- 1 \cdot x$ as $- x$ .

$y^{2} = - x + \frac{- 1}{- 1}$

Step 3.3.1.3

Divide $- 1$ by $- 1$ .

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

Step 4

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

$y = \pm \sqrt{- x + 1}$

Step 5

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

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Step 5.1

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

$y = \sqrt{- x + 1}$

Step 5.2

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

$y = - \sqrt{- x + 1}$

Step 5.3

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

$y = \sqrt{- x + 1}$

$y = - \sqrt{- x + 1}$

$y = \sqrt{- x + 1}$

$y = - \sqrt{- x + 1}$

Step 6

Set the radicand in $\sqrt{- x + 1}$ greater than or equal to $0$ to find where the expression is defined.

$- x + 1 \geq 0$

Step 7

Solve for $x$ .

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Step 7.1

Subtract $1$ from both sides of the inequality.

$- x \geq - 1$

Step 7.2

Divide each term in $- x \geq - 1$ by $- 1$ and simplify.

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Step 7.2.1

Divide each term in $- x \geq - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 1}{- 1}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 1}{- 1}$

Step 7.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 1}{- 1}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $- 1$ by $- 1$ .

$x \leq 1$

Step 8

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 1]$

Set-Builder Notation:

${x | x \leq 1}$

Step 9

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 10

Determine the domain and range.

Domain: $(- \infty, 1], {x | x \leq 1}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 11