Linear Algebra Examples

$- 2 (4 x + 3) = 3 (3 y + 4)$

Step 1

Rewrite the equation as $3 (3 y + 4) = - 2 (4 x + 3)$ .

$3 (3 y + 4) = - 2 (4 x + 3)$

Step 2

Divide each term in $3 (3 y + 4) = - 2 (4 x + 3)$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 (3 y + 4) = - 2 (4 x + 3)$ by $3$ .

$\frac{3 (3 y + 4)}{3} = \frac{- 2 (4 x + 3)}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 (3 y + 4)}{3} = \frac{- 2 (4 x + 3)}{3}$

Step 2.2.1.2

Divide $3 y + 4$ by $1$ .

$3 y + 4 = \frac{- 2 (4 x + 3)}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$3 y + 4 = - \frac{2 (4 x + 3)}{3}$

Step 3

Subtract $4$ from both sides of the equation.

$3 y = - \frac{2 (4 x + 3)}{3} - 4$

Step 4

Divide each term in $3 y = - \frac{2 (4 x + 3)}{3} - 4$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 y = - \frac{2 (4 x + 3)}{3} - 4$ by $3$ .

$\frac{3 y}{3} = \frac{- \frac{2 (4 x + 3)}{3}}{3} + \frac{- 4}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{- \frac{2 (4 x + 3)}{3}}{3} + \frac{- 4}{3}$

Step 4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{2 (4 x + 3)}{3}}{3} + \frac{- 4}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Combine the numerators over the common denominator.

$y = \frac{- \frac{2 (4 x + 3)}{3} - 4}{3}$

Step 4.3.2

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

$y = \frac{- \frac{2 (4 x + 3)}{3} - 4 \cdot \frac{3}{3}}{3}$

Step 4.3.3

Simplify terms.

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Step 4.3.3.1

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

$y = \frac{- \frac{2 (4 x + 3)}{3} + \frac{- 4 \cdot 3}{3}}{3}$

Step 4.3.3.2

Combine the numerators over the common denominator.

$y = \frac{\frac{- 2 (4 x + 3) - 4 \cdot 3}{3}}{3}$

Step 4.3.4

Simplify the numerator.

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Step 4.3.4.1

Factor $- 2$ out of $- 2 (4 x + 3) - 4 \cdot 3$ .

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Step 4.3.4.1.1

Factor $- 2$ out of $- 4 \cdot 3$ .

$y = \frac{\frac{- 2 (4 x + 3) - 2 (2 \cdot 3)}{3}}{3}$

Step 4.3.4.1.2

Factor $- 2$ out of $- 2 (4 x + 3) - 2 (2 \cdot 3)$ .

$y = \frac{\frac{- 2 (4 x + 3 + 2 \cdot 3)}{3}}{3}$

Step 4.3.4.2

Multiply $2$ by $3$ .

$y = \frac{\frac{- 2 (4 x + 3 + 6)}{3}}{3}$

Step 4.3.4.3

Add $3$ and $6$ .

$y = \frac{\frac{- 2 (4 x + 9)}{3}}{3}$

Step 4.3.5

Move the negative in front of the fraction.

$y = \frac{- \frac{2 (4 x + 9)}{3}}{3}$

Step 4.3.6

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{2 (4 x + 9)}{3} \cdot \frac{1}{3}$

Step 4.3.7

Multiply $- \frac{2 (4 x + 9)}{3} \cdot \frac{1}{3}$ .

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Step 4.3.7.1

Multiply $\frac{1}{3}$ by $\frac{2 (4 x + 9)}{3}$ .

$y = - \frac{2 (4 x + 9)}{3 \cdot 3}$

Step 4.3.7.2

Multiply $3$ by $3$ .

$y = - \frac{2 (4 x + 9)}{9}$

Step 5

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 6