Linear Algebra Examples

$- \frac{3}{5} x + \frac{4}{5} = \frac{5}{3} x - \frac{13}{3}$

Step 1

Simplify each term.

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Step 1.1

Combine $x$ and $\frac{3}{5}$ .

$- \frac{x \cdot 3}{5} + \frac{4}{5} = \frac{5}{3} x - \frac{13}{3}$

Step 1.2

Move $3$ to the left of $x$ .

$- \frac{3 x}{5} + \frac{4}{5} = \frac{5}{3} x - \frac{13}{3}$

Step 2

Combine $\frac{5}{3}$ and $x$ .

$- \frac{3 x}{5} + \frac{4}{5} = \frac{5 x}{3} - \frac{13}{3}$

Step 3

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

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Step 3.1

Subtract $\frac{5 x}{3}$ from both sides of the equation.

$- \frac{3 x}{5} + \frac{4}{5} - \frac{5 x}{3} = - \frac{13}{3}$

Step 3.2

To write $- \frac{3 x}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{3 x}{5} \cdot \frac{3}{3} - \frac{5 x}{3} + \frac{4}{5} = - \frac{13}{3}$

Step 3.3

To write $- \frac{5 x}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{3 x}{5} \cdot \frac{3}{3} - \frac{5 x}{3} \cdot \frac{5}{5} + \frac{4}{5} = - \frac{13}{3}$

Step 3.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{3 x}{5}$ by $\frac{3}{3}$ .

$- \frac{3 x \cdot 3}{5 \cdot 3} - \frac{5 x}{3} \cdot \frac{5}{5} + \frac{4}{5} = - \frac{13}{3}$

Step 3.4.2

Multiply $5$ by $3$ .

$- \frac{3 x \cdot 3}{15} - \frac{5 x}{3} \cdot \frac{5}{5} + \frac{4}{5} = - \frac{13}{3}$

Step 3.4.3

Multiply $\frac{5 x}{3}$ by $\frac{5}{5}$ .

$- \frac{3 x \cdot 3}{15} - \frac{5 x \cdot 5}{3 \cdot 5} + \frac{4}{5} = - \frac{13}{3}$

Step 3.4.4

Multiply $3$ by $5$ .

$- \frac{3 x \cdot 3}{15} - \frac{5 x \cdot 5}{15} + \frac{4}{5} = - \frac{13}{3}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{- 3 x \cdot 3 - 5 x \cdot 5}{15} + \frac{4}{5} = - \frac{13}{3}$

Step 3.6

Simplify each term.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Factor $x$ out of $- 3 x \cdot 3 - 5 x \cdot 5$ .

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Step 3.6.1.1.1

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

$\frac{x (- 3 \cdot 3) - 5 x \cdot 5}{15} + \frac{4}{5} = - \frac{13}{3}$

Step 3.6.1.1.2

Factor $x$ out of $- 5 x \cdot 5$ .

$\frac{x (- 3 \cdot 3) + x (- 5 \cdot 5)}{15} + \frac{4}{5} = - \frac{13}{3}$

Step 3.6.1.1.3

Factor $x$ out of $x (- 3 \cdot 3) + x (- 5 \cdot 5)$ .

$\frac{x (- 3 \cdot 3 - 5 \cdot 5)}{15} + \frac{4}{5} = - \frac{13}{3}$

Step 3.6.1.2

Multiply $- 3$ by $3$ .

$\frac{x (- 9 - 5 \cdot 5)}{15} + \frac{4}{5} = - \frac{13}{3}$

Step 3.6.1.3

Multiply $- 5$ by $5$ .

$\frac{x (- 9 - 25)}{15} + \frac{4}{5} = - \frac{13}{3}$

Step 3.6.1.4

Subtract $25$ from $- 9$ .

$\frac{x \cdot - 34}{15} + \frac{4}{5} = - \frac{13}{3}$

Step 3.6.2

Move $- 34$ to the left of $x$ .

$\frac{- 34 \cdot x}{15} + \frac{4}{5} = - \frac{13}{3}$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{34 x}{15} + \frac{4}{5} = - \frac{13}{3}$

Step 4

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

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Step 4.1

Subtract $\frac{4}{5}$ from both sides of the equation.

$- \frac{34 x}{15} = - \frac{13}{3} - \frac{4}{5}$

Step 4.2

To write $- \frac{13}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{34 x}{15} = - \frac{13}{3} \cdot \frac{5}{5} - \frac{4}{5}$

Step 4.3

To write $- \frac{4}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{34 x}{15} = - \frac{13}{3} \cdot \frac{5}{5} - \frac{4}{5} \cdot \frac{3}{3}$

Step 4.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

Multiply $\frac{13}{3}$ by $\frac{5}{5}$ .

$- \frac{34 x}{15} = - \frac{13 \cdot 5}{3 \cdot 5} - \frac{4}{5} \cdot \frac{3}{3}$

Step 4.4.2

Multiply $3$ by $5$ .

$- \frac{34 x}{15} = - \frac{13 \cdot 5}{15} - \frac{4}{5} \cdot \frac{3}{3}$

Step 4.4.3

Multiply $\frac{4}{5}$ by $\frac{3}{3}$ .

$- \frac{34 x}{15} = - \frac{13 \cdot 5}{15} - \frac{4 \cdot 3}{5 \cdot 3}$

Step 4.4.4

Multiply $5$ by $3$ .

$- \frac{34 x}{15} = - \frac{13 \cdot 5}{15} - \frac{4 \cdot 3}{15}$

Step 4.5

Combine the numerators over the common denominator.

$- \frac{34 x}{15} = \frac{- 13 \cdot 5 - 4 \cdot 3}{15}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 13$ by $5$ .

$- \frac{34 x}{15} = \frac{- 65 - 4 \cdot 3}{15}$

Step 4.6.2

Multiply $- 4$ by $3$ .

$- \frac{34 x}{15} = \frac{- 65 - 12}{15}$

Step 4.6.3

Subtract $12$ from $- 65$ .

$- \frac{34 x}{15} = \frac{- 77}{15}$

Step 4.7

Move the negative in front of the fraction.

$- \frac{34 x}{15} = - \frac{77}{15}$

Step 5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- 34 x = - 77$

Step 6

Divide each term in $- 34 x = - 77$ by $- 34$ and simplify.

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Step 6.1

Divide each term in $- 34 x = - 77$ by $- 34$ .

$\frac{- 34 x}{- 34} = \frac{- 77}{- 34}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 34$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 34 x}{- 34} = \frac{- 77}{- 34}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 77}{- 34}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Dividing two negative values results in a positive value.

$x = \frac{77}{34}$

Step 7

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

${x | x = \frac{77}{34}}$

Step 8