Algebra Examples

$2 x + 3 y - \frac{2}{3} x + 1 \frac{1}{2} y$

Step 1

Convert $1 \frac{1}{2}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$2 x + 3 y - \frac{2}{3} x + (1 + \frac{1}{2}) y$

Step 1.2

Add $1$ and $\frac{1}{2}$ .

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

$2 x + 3 y - \frac{2}{3} x + (\frac{2}{2} + \frac{1}{2}) y$

Step 1.2.2

Combine the numerators over the common denominator.

$2 x + 3 y - \frac{2}{3} x + \frac{2 + 1}{2} y$

Step 1.2.3

Add $2$ and $1$ .

$2 x + 3 y - \frac{2}{3} x + \frac{3}{2} y$

Step 2

Simplify each term.

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Step 2.1

Combine $x$ and $\frac{2}{3}$ .

$2 x + 3 y - \frac{x \cdot 2}{3} + \frac{3}{2} y$

Step 2.2

Move $2$ to the left of $x$ .

$2 x + 3 y - \frac{2 \cdot x}{3} + \frac{3}{2} y$

Step 2.3

Combine $\frac{3}{2}$ and $y$ .

$2 x + 3 y - \frac{2 x}{3} + \frac{3 y}{2}$

Step 3

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

$3 y + 2 x \cdot \frac{3}{3} - \frac{2 x}{3} + \frac{3 y}{2}$

Step 4

Simplify terms.

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Step 4.1

Combine $2 x$ and $\frac{3}{3}$ .

$3 y + \frac{2 x \cdot 3}{3} - \frac{2 x}{3} + \frac{3 y}{2}$

Step 4.2

Combine the numerators over the common denominator.

$3 y + \frac{2 x \cdot 3 - 2 x}{3} + \frac{3 y}{2}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Factor $2 x$ out of $- 2 x$ .

$3 y + \frac{2 x (3) + 2 x (- 1)}{3} + \frac{3 y}{2}$

Step 5.1.3

Factor $2 x$ out of $2 x (3) + 2 x (- 1)$ .

$3 y + \frac{2 x (3 - 1)}{3} + \frac{3 y}{2}$

Step 5.2

Subtract $1$ from $3$ .

$3 y + \frac{2 x \cdot 2}{3} + \frac{3 y}{2}$

Step 5.3

Multiply $2$ by $2$ .

$3 y + \frac{4 x}{3} + \frac{3 y}{2}$

Step 6

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

$3 y \cdot \frac{3}{3} + \frac{4 x}{3} + \frac{3 y}{2}$

Step 7

Combine $3 y$ and $\frac{3}{3}$ .

$\frac{3 y \cdot 3}{3} + \frac{4 x}{3} + \frac{3 y}{2}$

Step 8

Combine the numerators over the common denominator.

$\frac{3 y \cdot 3 + 4 x}{3} + \frac{3 y}{2}$

Step 9

Multiply $3$ by $3$ .

$\frac{9 y + 4 x}{3} + \frac{3 y}{2}$

Step 10

To write $\frac{9 y + 4 x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9 y + 4 x}{3} \cdot \frac{2}{2} + \frac{3 y}{2}$

Step 11

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

$\frac{9 y + 4 x}{3} \cdot \frac{2}{2} + \frac{3 y}{2} \cdot \frac{3}{3}$

Step 12

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

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Step 12.1

Multiply $\frac{9 y + 4 x}{3}$ by $\frac{2}{2}$ .

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

Step 12.2

Multiply $3$ by $2$ .

$\frac{(9 y + 4 x) \cdot 2}{6} + \frac{3 y}{2} \cdot \frac{3}{3}$

Step 12.3

Multiply $\frac{3 y}{2}$ by $\frac{3}{3}$ .

$\frac{(9 y + 4 x) \cdot 2}{6} + \frac{3 y \cdot 3}{2 \cdot 3}$

Step 12.4

Multiply $2$ by $3$ .

$\frac{(9 y + 4 x) \cdot 2}{6} + \frac{3 y \cdot 3}{6}$

Step 13

Combine the numerators over the common denominator.

$\frac{(9 y + 4 x) \cdot 2 + 3 y \cdot 3}{6}$

Step 14

Simplify the numerator.

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Step 14.1

Apply the distributive property.

$\frac{9 y \cdot 2 + 4 x \cdot 2 + 3 y \cdot 3}{6}$

Step 14.2

Multiply $2$ by $9$ .

$\frac{18 y + 4 x \cdot 2 + 3 y \cdot 3}{6}$

Step 14.3

Multiply $2$ by $4$ .

$\frac{18 y + 8 x + 3 y \cdot 3}{6}$

Step 14.4

Multiply $3$ by $3$ .

$\frac{18 y + 8 x + 9 y}{6}$

Step 14.5

Add $18 y$ and $9 y$ .

$\frac{8 x + 27 y}{6}$