Linear Algebra Examples

$4 - \frac{10 x + 1}{6} = \frac{4 x - 16 x + 3}{4}$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{4 x - 16 x + 3}{4} = 4 - \frac{10 x + 1}{6}$

Step 2

Multiply both sides by $4$ .

$\frac{4 x - 16 x + 3}{4} \cdot 4 = (4 - \frac{10 x + 1}{6}) \cdot 4$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{4 x - 16 x + 3}{4} \cdot 4$ .

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Step 3.1.1.1

Simplify the numerator.

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Step 3.1.1.1.1

Subtract $16 x$ from $4 x$ .

$\frac{- 12 x + 3}{4} \cdot 4 = (4 - \frac{10 x + 1}{6}) \cdot 4$

Step 3.1.1.1.2

Factor $3$ out of $- 12 x + 3$ .

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Step 3.1.1.1.2.1

Factor $3$ out of $- 12 x$ .

$\frac{3 (- 4 x) + 3}{4} \cdot 4 = (4 - \frac{10 x + 1}{6}) \cdot 4$

Step 3.1.1.1.2.2

Factor $3$ out of $3$ .

$\frac{3 (- 4 x) + 3 (1)}{4} \cdot 4 = (4 - \frac{10 x + 1}{6}) \cdot 4$

Step 3.1.1.1.2.3

Factor $3$ out of $3 (- 4 x) + 3 (1)$ .

$\frac{3 (- 4 x + 1)}{4} \cdot 4 = (4 - \frac{10 x + 1}{6}) \cdot 4$

Step 3.1.1.2

Simplify terms.

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Step 3.1.1.2.1

Cancel the common factor of $4$ .

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Step 3.1.1.2.1.1

Cancel the common factor.

$\frac{3 (- 4 x + 1)}{4} \cdot 4 = (4 - \frac{10 x + 1}{6}) \cdot 4$

Step 3.1.1.2.1.2

Rewrite the expression.

$3 (- 4 x + 1) = (4 - \frac{10 x + 1}{6}) \cdot 4$

Step 3.1.1.2.2

Apply the distributive property.

$3 (- 4 x) + 3 \cdot 1 = (4 - \frac{10 x + 1}{6}) \cdot 4$

Step 3.1.1.2.3

Multiply.

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Step 3.1.1.2.3.1

Multiply $- 4$ by $3$ .

$- 12 x + 3 \cdot 1 = (4 - \frac{10 x + 1}{6}) \cdot 4$

Step 3.1.1.2.3.2

Multiply $3$ by $1$ .

$- 12 x + 3 = (4 - \frac{10 x + 1}{6}) \cdot 4$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $(4 - \frac{10 x + 1}{6}) \cdot 4$ .

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Step 3.2.1.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- 12 x + 3 = (4 \cdot \frac{6}{6} - \frac{10 x + 1}{6}) \cdot 4$

Step 3.2.1.2

Simplify terms.

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Step 3.2.1.2.1

Combine $4$ and $\frac{6}{6}$ .

$- 12 x + 3 = (\frac{4 \cdot 6}{6} - \frac{10 x + 1}{6}) \cdot 4$

Step 3.2.1.2.2

Combine the numerators over the common denominator.

$- 12 x + 3 = \frac{4 \cdot 6 - (10 x + 1)}{6} \cdot 4$

Step 3.2.1.3

Simplify the numerator.

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Step 3.2.1.3.1

Multiply $4$ by $6$ .

$- 12 x + 3 = \frac{24 - (10 x + 1)}{6} \cdot 4$

Step 3.2.1.3.2

Apply the distributive property.

$- 12 x + 3 = \frac{24 - (10 x) - 1 \cdot 1}{6} \cdot 4$

Step 3.2.1.3.3

Multiply $10$ by $- 1$ .

$- 12 x + 3 = \frac{24 - 10 x - 1 \cdot 1}{6} \cdot 4$

Step 3.2.1.3.4

Multiply $- 1$ by $1$ .

$- 12 x + 3 = \frac{24 - 10 x - 1}{6} \cdot 4$

Step 3.2.1.3.5

Subtract $1$ from $24$ .

$- 12 x + 3 = \frac{- 10 x + 23}{6} \cdot 4$

Step 3.2.1.4

Simplify terms.

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Step 3.2.1.4.1

Cancel the common factor of $2$ .

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Step 3.2.1.4.1.1

Factor $2$ out of $6$ .

$- 12 x + 3 = \frac{- 10 x + 23}{2 (3)} \cdot 4$

Step 3.2.1.4.1.2

Factor $2$ out of $4$ .

$- 12 x + 3 = \frac{- 10 x + 23}{2 \cdot 3} \cdot (2 \cdot 2)$

Step 3.2.1.4.1.3

Cancel the common factor.

$- 12 x + 3 = \frac{- 10 x + 23}{2 \cdot 3} \cdot (2 \cdot 2)$

Step 3.2.1.4.1.4

Rewrite the expression.

$- 12 x + 3 = \frac{- 10 x + 23}{3} \cdot 2$

Step 3.2.1.4.2

Combine $\frac{- 10 x + 23}{3}$ and $2$ .

$- 12 x + 3 = \frac{(- 10 x + 23) \cdot 2}{3}$

Step 3.2.1.4.3

Move $2$ to the left of $- 10 x + 23$ .

$- 12 x + 3 = \frac{2 \cdot (- 10 x + 23)}{3}$

Step 3.2.1.4.4

Factor $- 1$ out of $- 10 x$ .

$- 12 x + 3 = \frac{2 (- (10 x) + 23)}{3}$

Step 3.2.1.4.5

Rewrite $23$ as $- 1 (- 23)$ .

$- 12 x + 3 = \frac{2 (- (10 x) - 1 (- 23))}{3}$

Step 3.2.1.4.6

Factor $- 1$ out of $- (10 x) - 1 (- 23)$ .

$- 12 x + 3 = \frac{2 (- (10 x - 23))}{3}$

Step 3.2.1.4.7

Simplify the expression.

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Step 3.2.1.4.7.1

Rewrite $- (10 x - 23)$ as $- 1 (10 x - 23)$ .

$- 12 x + 3 = \frac{2 (- 1 (10 x - 23))}{3}$

Step 3.2.1.4.7.2

Move the negative in front of the fraction.

$- 12 x + 3 = - \frac{2 (10 x - 23)}{3}$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides by $3$ .

$(- 12 x + 3) \cdot 3 = - \frac{2 (10 x - 23)}{3} \cdot 3$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $(- 12 x + 3) \cdot 3$ .

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Step 4.2.1.1.1

Apply the distributive property.

$- 12 x \cdot 3 + 3 \cdot 3 = - \frac{2 (10 x - 23)}{3} \cdot 3$

Step 4.2.1.1.2

Multiply.

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Step 4.2.1.1.2.1

Multiply $3$ by $- 12$ .

$- 36 x + 3 \cdot 3 = - \frac{2 (10 x - 23)}{3} \cdot 3$

Step 4.2.1.1.2.2

Multiply $3$ by $3$ .

$- 36 x + 9 = - \frac{2 (10 x - 23)}{3} \cdot 3$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $- \frac{2 (10 x - 23)}{3} \cdot 3$ .

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Step 4.2.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1.1

Move the leading negative in $- \frac{2 (10 x - 23)}{3}$ into the numerator.

$- 36 x + 9 = \frac{- 2 (10 x - 23)}{3} \cdot 3$

Step 4.2.2.1.1.2

Cancel the common factor.

$- 36 x + 9 = \frac{- 2 (10 x - 23)}{3} \cdot 3$

Step 4.2.2.1.1.3

Rewrite the expression.

$- 36 x + 9 = - 2 (10 x - 23)$

Step 4.2.2.1.2

Apply the distributive property.

$- 36 x + 9 = - 2 (10 x) - 2 \cdot - 23$

Step 4.2.2.1.3

Multiply.

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Step 4.2.2.1.3.1

Multiply $10$ by $- 2$ .

$- 36 x + 9 = - 20 x - 2 \cdot - 23$

Step 4.2.2.1.3.2

Multiply $- 2$ by $- 23$ .

$- 36 x + 9 = - 20 x + 46$

Step 4.3

Solve for $x$ .

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Step 4.3.1

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

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Step 4.3.1.1

Add $20 x$ to both sides of the equation.

$- 36 x + 9 + 20 x = 46$

Step 4.3.1.2

Add $- 36 x$ and $20 x$ .

$- 16 x + 9 = 46$

Step 4.3.2

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

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Step 4.3.2.1

Subtract $9$ from both sides of the equation.

$- 16 x = 46 - 9$

Step 4.3.2.2

Subtract $9$ from $46$ .

$- 16 x = 37$

Step 4.3.3

Divide each term in $- 16 x = 37$ by $- 16$ and simplify.

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Step 4.3.3.1

Divide each term in $- 16 x = 37$ by $- 16$ .

$\frac{- 16 x}{- 16} = \frac{37}{- 16}$

Step 4.3.3.2

Simplify the left side.

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Step 4.3.3.2.1

Cancel the common factor of $- 16$ .

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Step 4.3.3.2.1.1

Cancel the common factor.

$\frac{- 16 x}{- 16} = \frac{37}{- 16}$

Step 4.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{37}{- 16}$

Step 4.3.3.3

Simplify the right side.

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Step 4.3.3.3.1

Move the negative in front of the fraction.

$x = - \frac{37}{16}$

Step 5

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

${x | x = - \frac{37}{16}}$

Step 6