Linear Algebra Examples

$\frac{7}{8} x + \frac{1}{4} = \frac{3}{4} x$

Step 1

Combine $\frac{7}{8}$ and $x$ .

$\frac{7 x}{8} + \frac{1}{4} = \frac{3}{4} x$

Step 2

Combine $\frac{3}{4}$ and $x$ .

$\frac{7 x}{8} + \frac{1}{4} = \frac{3 x}{4}$

Step 3

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

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Step 3.1

Subtract $\frac{3 x}{4}$ from both sides of the equation.

$\frac{7 x}{8} + \frac{1}{4} - \frac{3 x}{4} = 0$

Step 3.2

To write $- \frac{3 x}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7 x}{8} - \frac{3 x}{4} \cdot \frac{2}{2} + \frac{1}{4} = 0$

Step 3.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{3 x}{4}$ by $\frac{2}{2}$ .

$\frac{7 x}{8} - \frac{3 x \cdot 2}{4 \cdot 2} + \frac{1}{4} = 0$

Step 3.3.2

Multiply $4$ by $2$ .

$\frac{7 x}{8} - \frac{3 x \cdot 2}{8} + \frac{1}{4} = 0$

Step 3.4

Combine the numerators over the common denominator.

$\frac{7 x - 3 x \cdot 2}{8} + \frac{1}{4} = 0$

Step 3.5

Simplify each term.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Factor $x$ out of $7 x - 3 x \cdot 2$ .

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Step 3.5.1.1.1

Factor $x$ out of $7 x$ .

$\frac{x \cdot 7 - 3 x \cdot 2}{8} + \frac{1}{4} = 0$

Step 3.5.1.1.2

Factor $x$ out of $- 3 x \cdot 2$ .

$\frac{x \cdot 7 + x (- 3 \cdot 2)}{8} + \frac{1}{4} = 0$

Step 3.5.1.1.3

Factor $x$ out of $x \cdot 7 + x (- 3 \cdot 2)$ .

$\frac{x (7 - 3 \cdot 2)}{8} + \frac{1}{4} = 0$

Step 3.5.1.2

Multiply $- 3$ by $2$ .

$\frac{x (7 - 6)}{8} + \frac{1}{4} = 0$

Step 3.5.1.3

Subtract $6$ from $7$ .

$\frac{x \cdot 1}{8} + \frac{1}{4} = 0$

Step 3.5.2

Multiply $x$ by $1$ .

$\frac{x}{8} + \frac{1}{4} = 0$

Step 4

Subtract $\frac{1}{4}$ from both sides of the equation.

$\frac{x}{8} = - \frac{1}{4}$

Step 5

Multiply both sides of the equation by $8$ .

$8 \frac{x}{8} = 8 (- \frac{1}{4})$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $8$ .

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Step 6.1.1.1

Cancel the common factor.

$8 \frac{x}{8} = 8 (- \frac{1}{4})$

Step 6.1.1.2

Rewrite the expression.

$x = 8 (- \frac{1}{4})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $8 (- \frac{1}{4})$ .

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Step 6.2.1.1

Cancel the common factor of $4$ .

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Step 6.2.1.1.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$x = 8 (\frac{- 1}{4})$

Step 6.2.1.1.2

Factor $4$ out of $8$ .

$x = 4 (2) \frac{- 1}{4}$

Step 6.2.1.1.3

Cancel the common factor.

$x = 4 \cdot 2 \frac{- 1}{4}$

Step 6.2.1.1.4

Rewrite the expression.

$x = 2 \cdot - 1$

Step 6.2.1.2

Multiply $2$ by $- 1$ .

$x = - 2$

Step 7

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

${x | x = - 2}$

Step 8