Algebra Examples

$\frac{2 x + 1}{6} - \frac{x - 3}{3} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 1

Simplify $\frac{2 x + 1}{6} - \frac{x - 3}{3}$ .

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Step 1.1

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

$\frac{2 x + 1}{6} - \frac{x - 3}{3} \cdot \frac{2}{2} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 1.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

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

$\frac{2 x + 1}{6} - \frac{(x - 3) \cdot 2}{3 \cdot 2} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 1.2.2

Multiply $3$ by $2$ .

$\frac{2 x + 1}{6} - \frac{(x - 3) \cdot 2}{6} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{2 x + 1 - (x - 3) \cdot 2}{6} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Apply the distributive property.

$\frac{2 x + 1 + (- x - - 3) \cdot 2}{6} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 1.4.2

Multiply $- 1$ by $- 3$ .

$\frac{2 x + 1 + (- x + 3) \cdot 2}{6} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 1.4.3

Apply the distributive property.

$\frac{2 x + 1 - x \cdot 2 + 3 \cdot 2}{6} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 1.4.4

Multiply $2$ by $- 1$ .

$\frac{2 x + 1 - 2 x + 3 \cdot 2}{6} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 1.4.5

Multiply $3$ by $2$ .

$\frac{2 x + 1 - 2 x + 6}{6} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 1.4.6

Subtract $2 x$ from $2 x$ .

$\frac{0 + 1 + 6}{6} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 1.4.7

Add $0$ and $1$ .

$\frac{1 + 6}{6} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 1.4.8

Add $1$ and $6$ .

$\frac{7}{6} = \frac{4 x - 1}{3} + \frac{x - 6}{2}$

Step 2

Simplify $\frac{4 x - 1}{3} + \frac{x - 6}{2}$ .

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Step 2.1

To write $\frac{4 x - 1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7}{6} = \frac{4 x - 1}{3} \cdot \frac{2}{2} + \frac{x - 6}{2}$

Step 2.2

To write $\frac{x - 6}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{7}{6} = \frac{4 x - 1}{3} \cdot \frac{2}{2} + \frac{x - 6}{2} \cdot \frac{3}{3}$

Step 2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{4 x - 1}{3}$ by $\frac{2}{2}$ .

$\frac{7}{6} = \frac{(4 x - 1) \cdot 2}{3 \cdot 2} + \frac{x - 6}{2} \cdot \frac{3}{3}$

Step 2.3.2

Multiply $3$ by $2$ .

$\frac{7}{6} = \frac{(4 x - 1) \cdot 2}{6} + \frac{x - 6}{2} \cdot \frac{3}{3}$

Step 2.3.3

Multiply $\frac{x - 6}{2}$ by $\frac{3}{3}$ .

$\frac{7}{6} = \frac{(4 x - 1) \cdot 2}{6} + \frac{(x - 6) \cdot 3}{2 \cdot 3}$

Step 2.3.4

Multiply $2$ by $3$ .

$\frac{7}{6} = \frac{(4 x - 1) \cdot 2}{6} + \frac{(x - 6) \cdot 3}{6}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{7}{6} = \frac{(4 x - 1) \cdot 2 + (x - 6) \cdot 3}{6}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{7}{6} = \frac{4 x \cdot 2 - 1 \cdot 2 + (x - 6) \cdot 3}{6}$

Step 2.5.2

Multiply $2$ by $4$ .

$\frac{7}{6} = \frac{8 x - 1 \cdot 2 + (x - 6) \cdot 3}{6}$

Step 2.5.3

Multiply $- 1$ by $2$ .

$\frac{7}{6} = \frac{8 x - 2 + (x - 6) \cdot 3}{6}$

Step 2.5.4

Apply the distributive property.

$\frac{7}{6} = \frac{8 x - 2 + x \cdot 3 - 6 \cdot 3}{6}$

Step 2.5.5

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

$\frac{7}{6} = \frac{8 x - 2 + 3 \cdot x - 6 \cdot 3}{6}$

Step 2.5.6

Multiply $- 6$ by $3$ .

$\frac{7}{6} = \frac{8 x - 2 + 3 \cdot x - 18}{6}$

Step 2.5.7

Add $8 x$ and $3 x$ .

$\frac{7}{6} = \frac{11 x - 2 - 18}{6}$

Step 2.5.8

Subtract $18$ from $- 2$ .

$\frac{7}{6} = \frac{11 x - 20}{6}$

Step 3

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

$\frac{11 x - 20}{6} = \frac{7}{6}$

Step 4

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

$11 x - 20 = 7$

Step 5

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

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Step 5.1

Add $20$ to both sides of the equation.

$11 x = 7 + 20$

Step 5.2

Add $7$ and $20$ .

$11 x = 27$

Step 6

Divide each term in $11 x = 27$ by $11$ and simplify.

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Step 6.1

Divide each term in $11 x = 27$ by $11$ .

$\frac{11 x}{11} = \frac{27}{11}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $11$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{27}{11}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{27}{11}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{27}{11}$

Decimal Form:

$x = 2.\overline{45}$

Mixed Number Form:

$x = 2 \frac{5}{11}$