Algebra Examples

$\frac{3}{4} (b - 4) + 7 = \frac{1}{3} b + 3 + \frac{3}{4} b$

Step 1

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

$\frac{1}{3} b + 3 + \frac{3}{4} b = \frac{3}{4} \cdot (b - 4) + 7$

Step 2

Simplify $\frac{1}{3} b + 3 + \frac{3}{4} b$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $\frac{1}{3}$ and $b$ .

$\frac{b}{3} + 3 + \frac{3}{4} b = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.1.2

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

$\frac{b}{3} + 3 + \frac{3 b}{4} = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.2

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

$\frac{b}{3} \cdot \frac{4}{4} + \frac{3 b}{4} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.3

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

$\frac{b}{3} \cdot \frac{4}{4} + \frac{3 b}{4} \cdot \frac{3}{3} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

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

$\frac{b \cdot 4}{3 \cdot 4} + \frac{3 b}{4} \cdot \frac{3}{3} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.4.2

Multiply $3$ by $4$ .

$\frac{b \cdot 4}{12} + \frac{3 b}{4} \cdot \frac{3}{3} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.4.3

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

$\frac{b \cdot 4}{12} + \frac{3 b \cdot 3}{4 \cdot 3} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.4.4

Multiply $4$ by $3$ .

$\frac{b \cdot 4}{12} + \frac{3 b \cdot 3}{12} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.5

Combine the numerators over the common denominator.

$\frac{b \cdot 4 + 3 b \cdot 3}{12} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.6

Simplify each term.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Factor $b$ out of $b \cdot 4 + 3 b \cdot 3$ .

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Step 2.6.1.1.1

Factor $b$ out of $b \cdot 4$ .

$\frac{b (4) + 3 b \cdot 3}{12} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.6.1.1.2

Factor $b$ out of $3 b \cdot 3$ .

$\frac{b (4) + b (3 \cdot 3)}{12} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.6.1.1.3

Factor $b$ out of $b (4) + b (3 \cdot 3)$ .

$\frac{b (4 + 3 \cdot 3)}{12} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.6.1.2

Multiply $3$ by $3$ .

$\frac{b (4 + 9)}{12} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.6.1.3

Add $4$ and $9$ .

$\frac{b \cdot 13}{12} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 2.6.2

Move $13$ to the left of $b$ .

$\frac{13 b}{12} + 3 = \frac{3}{4} \cdot (b - 4) + 7$

Step 3

Simplify $\frac{3}{4} \cdot (b - 4) + 7$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

$\frac{13 b}{12} + 3 = \frac{3}{4} b + \frac{3}{4} \cdot - 4 + 7$

Step 3.1.2

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

$\frac{13 b}{12} + 3 = \frac{3 b}{4} + \frac{3}{4} \cdot - 4 + 7$

Step 3.1.3

Cancel the common factor of $4$ .

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Step 3.1.3.1

Factor $4$ out of $- 4$ .

$\frac{13 b}{12} + 3 = \frac{3 b}{4} + \frac{3}{4} \cdot (4 (- 1)) + 7$

Step 3.1.3.2

Cancel the common factor.

$\frac{13 b}{12} + 3 = \frac{3 b}{4} + \frac{3}{4} \cdot (4 \cdot - 1) + 7$

Step 3.1.3.3

Rewrite the expression.

$\frac{13 b}{12} + 3 = \frac{3 b}{4} + 3 \cdot - 1 + 7$

Step 3.1.4

Multiply $3$ by $- 1$ .

$\frac{13 b}{12} + 3 = \frac{3 b}{4} - 3 + 7$

Step 3.2

Add $- 3$ and $7$ .

$\frac{13 b}{12} + 3 = \frac{3 b}{4} + 4$

Step 4

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

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Step 4.1

Subtract $\frac{3 b}{4}$ from both sides of the equation.

$\frac{13 b}{12} + 3 - \frac{3 b}{4} = 4$

Step 4.2

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

$\frac{13 b}{12} - \frac{3 b}{4} \cdot \frac{3}{3} + 3 = 4$

Step 4.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

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

$\frac{13 b}{12} - \frac{3 b \cdot 3}{4 \cdot 3} + 3 = 4$

Step 4.3.2

Multiply $4$ by $3$ .

$\frac{13 b}{12} - \frac{3 b \cdot 3}{12} + 3 = 4$

Step 4.4

Combine the numerators over the common denominator.

$\frac{13 b - 3 b \cdot 3}{12} + 3 = 4$

Step 4.5

Simplify each term.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Factor $b$ out of $13 b - 3 b \cdot 3$ .

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Step 4.5.1.1.1

Factor $b$ out of $13 b$ .

$\frac{b \cdot 13 - 3 b \cdot 3}{12} + 3 = 4$

Step 4.5.1.1.2

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

$\frac{b \cdot 13 + b (- 3 \cdot 3)}{12} + 3 = 4$

Step 4.5.1.1.3

Factor $b$ out of $b \cdot 13 + b (- 3 \cdot 3)$ .

$\frac{b (13 - 3 \cdot 3)}{12} + 3 = 4$

Step 4.5.1.2

Multiply $- 3$ by $3$ .

$\frac{b (13 - 9)}{12} + 3 = 4$

Step 4.5.1.3

Subtract $9$ from $13$ .

$\frac{b \cdot 4}{12} + 3 = 4$

Step 4.5.2

Cancel the common factor of $4$ and $12$ .

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Step 4.5.2.1

Factor $4$ out of $b \cdot 4$ .

$\frac{4 \cdot b}{12} + 3 = 4$

Step 4.5.2.2

Cancel the common factors.

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Step 4.5.2.2.1

Factor $4$ out of $12$ .

$\frac{4 \cdot b}{4 \cdot 3} + 3 = 4$

Step 4.5.2.2.2

Cancel the common factor.

$\frac{4 \cdot b}{4 \cdot 3} + 3 = 4$

Step 4.5.2.2.3

Rewrite the expression.

$\frac{b}{3} + 3 = 4$

Step 5

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

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Step 5.1

Subtract $3$ from both sides of the equation.

$\frac{b}{3} = 4 - 3$

Step 5.2

Subtract $3$ from $4$ .

$\frac{b}{3} = 1$

Step 6

Multiply both sides of the equation by $3$ .

$3 \frac{b}{3} = 3 \cdot 1$

Step 7

Simplify both sides of the equation.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Cancel the common factor of $3$ .

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Step 7.1.1.1

Cancel the common factor.

$3 \frac{b}{3} = 3 \cdot 1$

Step 7.1.1.2

Rewrite the expression.

$b = 3 \cdot 1$

Step 7.2

Simplify the right side.

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Step 7.2.1

Multiply $3$ by $1$ .

$b = 3$