Linear Algebra Examples

$5 x + \frac{7}{2} = \frac{3}{2} x - 14$

Step 1

Combine $\frac{3}{2}$ and $x$ .

$5 x + \frac{7}{2} = \frac{3 x}{2} - 14$

Step 2

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

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Step 2.1

Subtract $\frac{3 x}{2}$ from both sides of the equation.

$5 x + \frac{7}{2} - \frac{3 x}{2} = - 14$

Step 2.2

To write $5 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 x \cdot \frac{2}{2} - \frac{3 x}{2} + \frac{7}{2} = - 14$

Step 2.3

Combine $5 x$ and $\frac{2}{2}$ .

$\frac{5 x \cdot 2}{2} - \frac{3 x}{2} + \frac{7}{2} = - 14$

Step 2.4

Combine the numerators over the common denominator.

$\frac{5 x \cdot 2 - 3 x}{2} + \frac{7}{2} = - 14$

Step 2.5

Combine the numerators over the common denominator.

$\frac{5 x \cdot 2 - 3 x + 7}{2} = - 14$

Step 2.6

Multiply $2$ by $5$ .

$\frac{10 x - 3 x + 7}{2} = - 14$

Step 2.7

Subtract $3 x$ from $10 x$ .

$\frac{7 x + 7}{2} = - 14$

Step 2.8

Factor $7$ out of $7 x + 7$ .

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Step 2.8.1

Factor $7$ out of $7 x$ .

$\frac{7 (x) + 7}{2} = - 14$

Step 2.8.2

Factor $7$ out of $7$ .

$\frac{7 (x) + 7 (1)}{2} = - 14$

Step 2.8.3

Factor $7$ out of $7 (x) + 7 (1)$ .

$\frac{7 (x + 1)}{2} = - 14$

Step 3

Multiply both sides of the equation by $\frac{2}{7}$ .

$\frac{2}{7} \cdot \frac{7 (x + 1)}{2} = \frac{2}{7} \cdot - 14$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{2}{7} \cdot \frac{7 (x + 1)}{2}$ .

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Step 4.1.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{2}{7} \cdot \frac{7 (x + 1)}{2} = \frac{2}{7} \cdot - 14$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 (x + 1)) = \frac{2}{7} \cdot - 14$

Step 4.1.1.2

Cancel the common factor of $7$ .

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Step 4.1.1.2.1

Cancel the common factor.

$\frac{1}{7} (7 (x + 1)) = \frac{2}{7} \cdot - 14$

Step 4.1.1.2.2

Rewrite the expression.

$x + 1 = \frac{2}{7} \cdot - 14$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{2}{7} \cdot - 14$ .

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Step 4.2.1.1

Cancel the common factor of $7$ .

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Step 4.2.1.1.1

Factor $7$ out of $- 14$ .

$x + 1 = \frac{2}{7} \cdot (7 (- 2))$

Step 4.2.1.1.2

Cancel the common factor.

$x + 1 = \frac{2}{7} \cdot (7 \cdot - 2)$

Step 4.2.1.1.3

Rewrite the expression.

$x + 1 = 2 \cdot - 2$

Step 4.2.1.2

Multiply $2$ by $- 2$ .

$x + 1 = - 4$

Step 5

Move all terms not containing $x$ to the right side of the equation.

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Step 5.1

Subtract $1$ from both sides of the equation.

$x = - 4 - 1$

Step 5.2

Subtract $1$ from $- 4$ .

$x = - 5$

Step 6

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

${x | x = - 5}$

Step 7