Linear Algebra Examples

$x^{2} + y^{2} = 13$

Step 1

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

$y^{2} = 13 - x^{2}$

Step 2

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

$y = \pm \sqrt{13 - x^{2}}$

Step 3

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

Tap for more steps...

Step 3.1

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

$y = \sqrt{13 - x^{2}}$

Step 3.2

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

$y = - \sqrt{13 - x^{2}}$

Step 3.3

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

$y = \sqrt{13 - x^{2}}$

$y = - \sqrt{13 - x^{2}}$

$y = \sqrt{13 - x^{2}}$

$y = - \sqrt{13 - x^{2}}$

Step 4

Set the radicand in $\sqrt{13 - x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$13 - x^{2} \geq 0$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

Subtract $13$ from both sides of the inequality.

$- x^{2} \geq - 13$

Step 5.2

Divide each term in $- x^{2} \geq - 13$ by $- 1$ and simplify.

Tap for more steps...

Step 5.2.1

Divide each term in $- x^{2} \geq - 13$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 13}{- 1}$

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 13}{- 1}$

Step 5.2.2.2

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

$x^{2} \leq \frac{- 13}{- 1}$

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Divide $- 13$ by $- 1$ .

$x^{2} \leq 13$

Step 5.3

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

$\sqrt{x^{2}} \leq \sqrt{13}$

Step 5.4

Simplify the left side.

Tap for more steps...

Step 5.4.1

Pull terms out from under the radical.

$| x | \leq \sqrt{13}$

Step 5.5

Write $| x | \leq \sqrt{13}$ as a piecewise.

Tap for more steps...

Step 5.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 5.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq \sqrt{13}$

Step 5.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 5.5.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq \sqrt{13}$

Step 5.5.5

Write as a piecewise.

${\begin{cases} x \leq \sqrt{13} & x \geq 0 \\ - x \leq \sqrt{13} & x < 0 \end{cases}$

Step 5.6

Find the intersection of $x \leq \sqrt{13}$ and $x \geq 0$ .

$0 \leq x \leq \sqrt{13}$

Step 5.7

Solve $- x \leq \sqrt{13}$ when $x < 0$ .

Tap for more steps...

Step 5.7.1

Divide each term in $- x \leq \sqrt{13}$ by $- 1$ and simplify.

Tap for more steps...

Step 5.7.1.1

Divide each term in $- x \leq \sqrt{13}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{\sqrt{13}}{- 1}$

Step 5.7.1.2

Simplify the left side.

Tap for more steps...

Step 5.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{\sqrt{13}}{- 1}$

Step 5.7.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{\sqrt{13}}{- 1}$

Step 5.7.1.3

Simplify the right side.

Tap for more steps...

Step 5.7.1.3.1

Move the negative one from the denominator of $\frac{\sqrt{13}}{- 1}$ .

$x \geq - 1 \cdot \sqrt{13}$

Step 5.7.1.3.2

Rewrite $- 1 \cdot \sqrt{13}$ as $- \sqrt{13}$ .

$x \geq - \sqrt{13}$

Step 5.7.2

Find the intersection of $x \geq - \sqrt{13}$ and $x < 0$ .

$- \sqrt{13} \leq x < 0$

Step 5.8

Find the union of the solutions.

$- \sqrt{13} \leq x \leq \sqrt{13}$

Step 6

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

Interval Notation:

$[- \sqrt{13}, \sqrt{13}]$

Set-Builder Notation:

${x | - \sqrt{13} \leq x \leq \sqrt{13}}$

Step 7