Linear Algebra Examples

$\frac{x^{2} + 6 x}{10} = \frac{x}{5}$

Step 1

Multiply both sides by $10$ .

$\frac{x^{2} + 6 x}{10} \cdot 10 = \frac{x}{5} \cdot 10$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{x^{2} + 6 x}{10} \cdot 10$ .

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Step 2.1.1.1

Factor $x$ out of $x^{2} + 6 x$ .

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Step 2.1.1.1.1

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

$\frac{x \cdot x + 6 x}{10} \cdot 10 = \frac{x}{5} \cdot 10$

Step 2.1.1.1.2

Factor $x$ out of $6 x$ .

$\frac{x \cdot x + x \cdot 6}{10} \cdot 10 = \frac{x}{5} \cdot 10$

Step 2.1.1.1.3

Factor $x$ out of $x \cdot x + x \cdot 6$ .

$\frac{x (x + 6)}{10} \cdot 10 = \frac{x}{5} \cdot 10$

Step 2.1.1.2

Cancel the common factor of $10$ .

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Step 2.1.1.2.1

Cancel the common factor.

$\frac{x (x + 6)}{10} \cdot 10 = \frac{x}{5} \cdot 10$

Step 2.1.1.2.2

Rewrite the expression.

$x (x + 6) = \frac{x}{5} \cdot 10$

Step 2.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot 6 = \frac{x}{5} \cdot 10$

Step 2.1.1.4

Simplify the expression.

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Step 2.1.1.4.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 6 = \frac{x}{5} \cdot 10$

Step 2.1.1.4.2

Move $6$ to the left of $x$ .

$x^{2} + 6 x = \frac{x}{5} \cdot 10$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{x}{5} \cdot 10$ .

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Step 2.2.1.1

Cancel the common factor of $5$ .

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Step 2.2.1.1.1

Factor $5$ out of $10$ .

$x^{2} + 6 x = \frac{x}{5} \cdot (5 (2))$

Step 2.2.1.1.2

Cancel the common factor.

$x^{2} + 6 x = \frac{x}{5} \cdot (5 \cdot 2)$

Step 2.2.1.1.3

Rewrite the expression.

$x^{2} + 6 x = x \cdot 2$

Step 2.2.1.2

Move $2$ to the left of $x$ .

$x^{2} + 6 x = 2 x$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $2 x$ from both sides of the equation.

$x^{2} + 6 x - 2 x = 0$

Step 3.1.2

Subtract $2 x$ from $6 x$ .

$x^{2} + 4 x = 0$

Step 3.2

Factor $x$ out of $x^{2} + 4 x$ .

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Step 3.2.1

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

$x \cdot x + 4 x = 0$

Step 3.2.2

Factor $x$ out of $4 x$ .

$x \cdot x + x \cdot 4 = 0$

Step 3.2.3

Factor $x$ out of $x \cdot x + x \cdot 4$ .

$x (x + 4) = 0$

Step 3.3

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

$x = 0$

$x + 4 = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 3.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 3.6

The final solution is all the values that make $x (x + 4) = 0$ true.

$x = 0, - 4$

Step 4

The domain is the set of all valid $x$ values.

${x | x = 0, - 4}$

Step 5