Algebra Examples

$\frac{5}{3} t = 4 (\frac{1}{2} t + 2) - 2 (6 - \frac{3}{2} t)$

Step 1

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

$4 (\frac{1}{2} t + 2) - 2 (6 - \frac{3}{2} t) = \frac{5}{3} t$

Step 2

Simplify $4 (\frac{1}{2} t + 2) - 2 (6 - \frac{3}{2} t)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $\frac{1}{2}$ and $t$ .

$4 (\frac{t}{2} + 2) - 2 (6 - \frac{3}{2} t) = \frac{5}{3} t$

Step 2.1.2

Apply the distributive property.

$4 \frac{t}{2} + 4 \cdot 2 - 2 (6 - \frac{3}{2} t) = \frac{5}{3} t$

Step 2.1.3

Cancel the common factor of $2$ .

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Step 2.1.3.1

Factor $2$ out of $4$ .

$2 (2) \frac{t}{2} + 4 \cdot 2 - 2 (6 - \frac{3}{2} t) = \frac{5}{3} t$

Step 2.1.3.2

Cancel the common factor.

$2 \cdot 2 \frac{t}{2} + 4 \cdot 2 - 2 (6 - \frac{3}{2} t) = \frac{5}{3} t$

Step 2.1.3.3

Rewrite the expression.

$2 t + 4 \cdot 2 - 2 (6 - \frac{3}{2} t) = \frac{5}{3} t$

Step 2.1.4

Multiply $4$ by $2$ .

$2 t + 8 - 2 (6 - \frac{3}{2} t) = \frac{5}{3} t$

Step 2.1.5

Simplify each term.

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Step 2.1.5.1

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

$2 t + 8 - 2 (6 - \frac{t \cdot 3}{2}) = \frac{5}{3} t$

Step 2.1.5.2

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

$2 t + 8 - 2 (6 - \frac{3 t}{2}) = \frac{5}{3} t$

Step 2.1.6

Apply the distributive property.

$2 t + 8 - 2 \cdot 6 - 2 (- \frac{3 t}{2}) = \frac{5}{3} t$

Step 2.1.7

Multiply $- 2$ by $6$ .

$2 t + 8 - 12 - 2 (- \frac{3 t}{2}) = \frac{5}{3} t$

Step 2.1.8

Cancel the common factor of $2$ .

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Step 2.1.8.1

Move the leading negative in $- \frac{3 t}{2}$ into the numerator.

$2 t + 8 - 12 - 2 \frac{- 3 t}{2} = \frac{5}{3} t$

Step 2.1.8.2

Factor $2$ out of $- 2$ .

$2 t + 8 - 12 + 2 (- 1) \frac{- 3 t}{2} = \frac{5}{3} t$

Step 2.1.8.3

Cancel the common factor.

$2 t + 8 - 12 + 2 \cdot - 1 \frac{- 3 t}{2} = \frac{5}{3} t$

Step 2.1.8.4

Rewrite the expression.

$2 t + 8 - 12 - 1 (- 3 t) = \frac{5}{3} t$

Step 2.1.9

Multiply $- 3$ by $- 1$ .

$2 t + 8 - 12 + 3 t = \frac{5}{3} t$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $2 t$ and $3 t$ .

$5 t + 8 - 12 = \frac{5}{3} t$

Step 2.2.2

Subtract $12$ from $8$ .

$5 t - 4 = \frac{5}{3} t$

Step 3

Combine $\frac{5}{3}$ and $t$ .

$5 t - 4 = \frac{5 t}{3}$

Step 4

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

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Step 4.1

Subtract $\frac{5 t}{3}$ from both sides of the equation.

$5 t - 4 - \frac{5 t}{3} = 0$

Step 4.2

To write $5 t$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$5 t \cdot \frac{3}{3} - \frac{5 t}{3} - 4 = 0$

Step 4.3

Combine $5 t$ and $\frac{3}{3}$ .

$\frac{5 t \cdot 3}{3} - \frac{5 t}{3} - 4 = 0$

Step 4.4

Combine the numerators over the common denominator.

$\frac{5 t \cdot 3 - 5 t}{3} - 4 = 0$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Factor $5 t$ out of $5 t \cdot 3 - 5 t$ .

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Step 4.5.1.1

Factor $5 t$ out of $5 t \cdot 3$ .

$\frac{5 t (3) - 5 t}{3} - 4 = 0$

Step 4.5.1.2

Factor $5 t$ out of $- 5 t$ .

$\frac{5 t (3) + 5 t (- 1)}{3} - 4 = 0$

Step 4.5.1.3

Factor $5 t$ out of $5 t (3) + 5 t (- 1)$ .

$\frac{5 t (3 - 1)}{3} - 4 = 0$

Step 4.5.2

Subtract $1$ from $3$ .

$\frac{5 t \cdot 2}{3} - 4 = 0$

Step 4.5.3

Multiply $2$ by $5$ .

$\frac{10 t}{3} - 4 = 0$

Step 5

Add $4$ to both sides of the equation.

$\frac{10 t}{3} = 4$

Step 6

Multiply both sides of the equation by $\frac{3}{10}$ .

$\frac{3}{10} \cdot \frac{10 t}{3} = \frac{3}{10} \cdot 4$

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

Simplify $\frac{3}{10} \cdot \frac{10 t}{3}$ .

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Step 7.1.1.1

Cancel the common factor of $3$ .

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Step 7.1.1.1.1

Cancel the common factor.

$\frac{3}{10} \cdot \frac{10 t}{3} = \frac{3}{10} \cdot 4$

Step 7.1.1.1.2

Rewrite the expression.

$\frac{1}{10} (10 t) = \frac{3}{10} \cdot 4$

Step 7.1.1.2

Cancel the common factor of $10$ .

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Step 7.1.1.2.1

Factor $10$ out of $10 t$ .

$\frac{1}{10} (10 (t)) = \frac{3}{10} \cdot 4$

Step 7.1.1.2.2

Cancel the common factor.

$\frac{1}{10} (10 t) = \frac{3}{10} \cdot 4$

Step 7.1.1.2.3

Rewrite the expression.

$t = \frac{3}{10} \cdot 4$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{3}{10} \cdot 4$ .

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Step 7.2.1.1

Cancel the common factor of $2$ .

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Step 7.2.1.1.1

Factor $2$ out of $10$ .

$t = \frac{3}{2 (5)} \cdot 4$

Step 7.2.1.1.2

Factor $2$ out of $4$ .

$t = \frac{3}{2 \cdot 5} \cdot (2 \cdot 2)$

Step 7.2.1.1.3

Cancel the common factor.

$t = \frac{3}{2 \cdot 5} \cdot (2 \cdot 2)$

Step 7.2.1.1.4

Rewrite the expression.

$t = \frac{3}{5} \cdot 2$

Step 7.2.1.2

Combine $\frac{3}{5}$ and $2$ .

$t = \frac{3 \cdot 2}{5}$

Step 7.2.1.3

Multiply $3$ by $2$ .

$t = \frac{6}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$t = \frac{6}{5}$

Decimal Form:

$t = 1.2$

Mixed Number Form:

$t = 1 \frac{1}{5}$