Linear Algebra Examples

$x u \cdot u x + y u \cdot u y = 3 u$

Step 1

Simplify each term.

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Step 1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.1

Move $x$ .

$x \cdot x u \cdot u + y u \cdot u y = 3 u$

Step 1.1.2

Multiply $x$ by $x$ .

$x^{2} u \cdot u + y u \cdot u y = 3 u$

Step 1.2

Multiply $u$ by $u$ by adding the exponents.

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Step 1.2.1

Move $u$ .

$x^{2} (u \cdot u) + y u \cdot u y = 3 u$

Step 1.2.2

Multiply $u$ by $u$ .

$x^{2} u^{2} + y u \cdot u y = 3 u$

Step 1.3

Multiply $y$ by $y$ by adding the exponents.

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Step 1.3.1

Move $y$ .

$x^{2} u^{2} + y \cdot y u \cdot u = 3 u$

Step 1.3.2

Multiply $y$ by $y$ .

$x^{2} u^{2} + y^{2} u \cdot u = 3 u$

Step 1.4

Multiply $u$ by $u$ by adding the exponents.

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Step 1.4.1

Move $u$ .

$x^{2} u^{2} + y^{2} (u \cdot u) = 3 u$

Step 1.4.2

Multiply $u$ by $u$ .

$x^{2} u^{2} + y^{2} u^{2} = 3 u$

Step 2

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

$y^{2} u^{2} = 3 u - x^{2} u^{2}$

Step 3

Divide each term in $y^{2} u^{2} = 3 u - x^{2} u^{2}$ by $u^{2}$ and simplify.

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Step 3.1

Divide each term in $y^{2} u^{2} = 3 u - x^{2} u^{2}$ by $u^{2}$ .

$\frac{y^{2} u^{2}}{u^{2}} = \frac{3 u}{u^{2}} + \frac{- x^{2} u^{2}}{u^{2}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $u^{2}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{y^{2} u^{2}}{u^{2}} = \frac{3 u}{u^{2}} + \frac{- x^{2} u^{2}}{u^{2}}$

Step 3.2.1.2

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

$y^{2} = \frac{3 u}{u^{2}} + \frac{- x^{2} u^{2}}{u^{2}}$

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

Cancel the common factor of $u$ and $u^{2}$ .

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Step 3.3.1.1.1

Factor $u$ out of $3 u$ .

$y^{2} = \frac{u \cdot 3}{u^{2}} + \frac{- x^{2} u^{2}}{u^{2}}$

Step 3.3.1.1.2

Cancel the common factors.

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Step 3.3.1.1.2.1

Factor $u$ out of $u^{2}$ .

$y^{2} = \frac{u \cdot 3}{u \cdot u} + \frac{- x^{2} u^{2}}{u^{2}}$

Step 3.3.1.1.2.2

Cancel the common factor.

$y^{2} = \frac{u \cdot 3}{u \cdot u} + \frac{- x^{2} u^{2}}{u^{2}}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y^{2} = \frac{3}{u} + \frac{- x^{2} u^{2}}{u^{2}}$

Step 3.3.1.2

Cancel the common factor of $u^{2}$ .

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Step 3.3.1.2.1

Cancel the common factor.

$y^{2} = \frac{3}{u} + \frac{- x^{2} u^{2}}{u^{2}}$

Step 3.3.1.2.2

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

$y^{2} = \frac{3}{u} - x^{2}$

Step 4

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

$y = \pm \sqrt{\frac{3}{u} - x^{2}}$

Step 5

Simplify $\pm \sqrt{\frac{3}{u} - x^{2}}$ .

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Step 5.1

To write $- x^{2}$ as a fraction with a common denominator, multiply by $\frac{u}{u}$ .

$y = \pm \sqrt{\frac{3}{u} - x^{2} \cdot \frac{u}{u}}$

Step 5.2

Combine $- x^{2}$ and $\frac{u}{u}$ .

$y = \pm \sqrt{\frac{3}{u} + \frac{- x^{2} u}{u}}$

Step 5.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{3 - x^{2} u}{u}}$

Step 5.4

Rewrite $\sqrt{\frac{3 - x^{2} u}{u}}$ as $\frac{\sqrt{3 - x^{2} u}}{\sqrt{u}}$ .

$y = \pm \frac{\sqrt{3 - x^{2} u}}{\sqrt{u}}$

Step 5.5

Multiply $\frac{\sqrt{3 - x^{2} u}}{\sqrt{u}}$ by $\frac{\sqrt{u}}{\sqrt{u}}$ .

$y = \pm \frac{\sqrt{3 - x^{2} u}}{\sqrt{u}} \cdot \frac{\sqrt{u}}{\sqrt{u}}$

Step 5.6

Combine and simplify the denominator.

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Step 5.6.1

Multiply $\frac{\sqrt{3 - x^{2} u}}{\sqrt{u}}$ by $\frac{\sqrt{u}}{\sqrt{u}}$ .

$y = \pm \frac{\sqrt{3 - x^{2} u} \sqrt{u}}{\sqrt{u} \sqrt{u}}$

Step 5.6.2

Raise $\sqrt{u}$ to the power of $1$ .

$y = \pm \frac{\sqrt{3 - x^{2} u} \sqrt{u}}{{\sqrt{u}}^{1} \sqrt{u}}$

Step 5.6.3

Raise $\sqrt{u}$ to the power of $1$ .

$y = \pm \frac{\sqrt{3 - x^{2} u} \sqrt{u}}{{\sqrt{u}}^{1} {\sqrt{u}}^{1}}$

Step 5.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{3 - x^{2} u} \sqrt{u}}{{\sqrt{u}}^{1 + 1}}$

Step 5.6.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{3 - x^{2} u} \sqrt{u}}{{\sqrt{u}}^{2}}$

Step 5.6.6

Rewrite ${\sqrt{u}}^{2}$ as $u$ .

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Step 5.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{3 - x^{2} u} \sqrt{u}}{{(u^{\frac{1}{2}})}^{2}}$

Step 5.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm \frac{\sqrt{3 - x^{2} u} \sqrt{u}}{u^{\frac{1}{2} \cdot 2}}$

Step 5.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{\sqrt{3 - x^{2} u} \sqrt{u}}{u^{\frac{2}{2}}}$

Step 5.6.6.4

Cancel the common factor of $2$ .

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Step 5.6.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{3 - x^{2} u} \sqrt{u}}{u^{\frac{2}{2}}}$

Step 5.6.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{3 - x^{2} u} \sqrt{u}}{u^{1}}$

Step 5.6.6.5

Simplify.

$y = \pm \frac{\sqrt{3 - x^{2} u} \sqrt{u}}{u}$

Step 5.7

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(3 - x^{2} u) u}}{u}$

Step 5.8

Reorder factors in $\pm \frac{\sqrt{(3 - x^{2} u) u}}{u}$ .

$y = \pm \frac{\sqrt{u (3 - x^{2} u)}}{u}$

Step 6

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

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Step 6.1

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

$y = \frac{\sqrt{u (3 - x^{2} u)}}{u}$

Step 6.2

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

$y = - \frac{\sqrt{u (3 - x^{2} u)}}{u}$

Step 6.3

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

$y = \frac{\sqrt{u (3 - x^{2} u)}}{u}$

$y = - \frac{\sqrt{u (3 - x^{2} u)}}{u}$

$y = \frac{\sqrt{u (3 - x^{2} u)}}{u}$

$y = - \frac{\sqrt{u (3 - x^{2} u)}}{u}$

Step 7

Set the radicand in $\sqrt{u (3 - x^{2} u)}$ greater than or equal to $0$ to find where the expression is defined.

$u (3 - x^{2} u) \geq 0$

Step 8

Solve for $u$ .

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Step 8.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u = 0$

$3 - x^{2} u = 0$

Step 8.2

Set $u$ equal to $0$ .

$u = 0$

Step 8.3

Set $3 - x^{2} u$ equal to $0$ and solve for $u$ .

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Step 8.3.1

Set $3 - x^{2} u$ equal to $0$ .

$3 - x^{2} u = 0$

Step 8.3.2

Solve $3 - x^{2} u = 0$ for $u$ .

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Step 8.3.2.1

Subtract $3$ from both sides of the equation.

$- x^{2} u = - 3$

Step 8.3.2.2

Divide each term in $- x^{2} u = - 3$ by $- x^{2}$ and simplify.

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Step 8.3.2.2.1

Divide each term in $- x^{2} u = - 3$ by $- x^{2}$ .

$\frac{- x^{2} u}{- x^{2}} = \frac{- 3}{- x^{2}}$

Step 8.3.2.2.2

Simplify the left side.

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Step 8.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2} u}{x^{2}} = \frac{- 3}{- x^{2}}$

Step 8.3.2.2.2.2

Cancel the common factor of $x^{2}$ .

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Step 8.3.2.2.2.2.1

Cancel the common factor.

$\frac{x^{2} u}{x^{2}} = \frac{- 3}{- x^{2}}$

Step 8.3.2.2.2.2.2

Divide $u$ by $1$ .

$u = \frac{- 3}{- x^{2}}$

Step 8.3.2.2.3

Simplify the right side.

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Step 8.3.2.2.3.1

Dividing two negative values results in a positive value.

$u = \frac{3}{x^{2}}$

Step 8.4

The final solution is all the values that make $u (3 - x^{2} u) \geq 0$ true.

$u = 0$

$u = \frac{3}{x^{2}}$

$u = 0$

$u = \frac{3}{x^{2}}$

Step 9

Set the denominator in $\frac{\sqrt{u (3 - x^{2} u)}}{u}$ equal to $0$ to find where the expression is undefined.

$u = 0$

Step 10

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

$(N o (Min i m u m), N o (Max i m u m)]$

Set-Builder Notation:

${u | N o (Min i m u m) < u \leq N o (Max i m u m)}$