Linear Algebra Examples

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

Step 1

Rewrite the equation as $x^{2} + y^{2} = z$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

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

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

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

Step 5

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

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

Step 6

Add $x^{2}$ to both sides of the inequality.

$z \geq x^{2}$

Step 7

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${z | z \in ℝ}$