Algebra Examples

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

Step 1

Simplify $\frac{t + 5}{8} - \frac{t - 2}{2}$ .

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Step 1.1

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

$\frac{t + 5}{8} - \frac{t - 2}{2} \cdot \frac{4}{4} = \frac{1}{3}$

Step 1.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

Multiply $\frac{t - 2}{2}$ by $\frac{4}{4}$ .

$\frac{t + 5}{8} - \frac{(t - 2) \cdot 4}{2 \cdot 4} = \frac{1}{3}$

Step 1.2.2

Multiply $2$ by $4$ .

$\frac{t + 5}{8} - \frac{(t - 2) \cdot 4}{8} = \frac{1}{3}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{t + 5 - (t - 2) \cdot 4}{8} = \frac{1}{3}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Apply the distributive property.

$\frac{t + 5 + (- t - - 2) \cdot 4}{8} = \frac{1}{3}$

Step 1.4.2

Multiply $- 1$ by $- 2$ .

$\frac{t + 5 + (- t + 2) \cdot 4}{8} = \frac{1}{3}$

Step 1.4.3

Apply the distributive property.

$\frac{t + 5 - t \cdot 4 + 2 \cdot 4}{8} = \frac{1}{3}$

Step 1.4.4

Multiply $4$ by $- 1$ .

$\frac{t + 5 - 4 t + 2 \cdot 4}{8} = \frac{1}{3}$

Step 1.4.5

Multiply $2$ by $4$ .

$\frac{t + 5 - 4 t + 8}{8} = \frac{1}{3}$

Step 1.4.6

Subtract $4 t$ from $t$ .

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

Step 1.4.7

Add $5$ and $8$ .

$\frac{- 3 t + 13}{8} = \frac{1}{3}$

Step 1.5

Simplify with factoring out.

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Step 1.5.1

Factor $- 1$ out of $- 3 t$ .

$\frac{- (3 t) + 13}{8} = \frac{1}{3}$

Step 1.5.2

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{- (3 t) - 1 (- 13)}{8} = \frac{1}{3}$

Step 1.5.3

Factor $- 1$ out of $- (3 t) - 1 (- 13)$ .

$\frac{- (3 t - 13)}{8} = \frac{1}{3}$

Step 1.5.4

Simplify the expression.

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Step 1.5.4.1

Rewrite $- (3 t - 13)$ as $- 1 (3 t - 13)$ .

$\frac{- 1 (3 t - 13)}{8} = \frac{1}{3}$

Step 1.5.4.2

Move the negative in front of the fraction.

$- \frac{3 t - 13}{8} = \frac{1}{3}$

Step 2

Multiply both sides of the equation by $- 8$ .

$- 8 (- \frac{3 t - 13}{8}) = - 8 (\frac{1}{3})$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $- 8 (- \frac{3 t - 13}{8})$ .

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Step 3.1.1.1

Cancel the common factor of $8$ .

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Step 3.1.1.1.1

Move the leading negative in $- \frac{3 t - 13}{8}$ into the numerator.

$- 8 \frac{- (3 t - 13)}{8} = - 8 (\frac{1}{3})$

Step 3.1.1.1.2

Factor $8$ out of $- 8$ .

$8 (- 1) \frac{- (3 t - 13)}{8} = - 8 (\frac{1}{3})$

Step 3.1.1.1.3

Cancel the common factor.

$8 \cdot - 1 \frac{- (3 t - 13)}{8} = - 8 (\frac{1}{3})$

Step 3.1.1.1.4

Rewrite the expression.

$- - (3 t - 13) = - 8 (\frac{1}{3})$

Step 3.1.1.2

Multiply.

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Step 3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (3 t - 13) = - 8 (\frac{1}{3})$

Step 3.1.1.2.2

Multiply $3 t - 13$ by $1$ .

$3 t - 13 = - 8 (\frac{1}{3})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- 8 (\frac{1}{3})$ .

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Step 3.2.1.1

Combine $- 8$ and $\frac{1}{3}$ .

$3 t - 13 = \frac{- 8}{3}$

Step 3.2.1.2

Move the negative in front of the fraction.

$3 t - 13 = - \frac{8}{3}$

Step 4

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

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Step 4.1

Add $13$ to both sides of the equation.

$3 t = - \frac{8}{3} + 13$

Step 4.2

To write $13$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 t = - \frac{8}{3} + 13 \cdot \frac{3}{3}$

Step 4.3

Combine $13$ and $\frac{3}{3}$ .

$3 t = - \frac{8}{3} + \frac{13 \cdot 3}{3}$

Step 4.4

Combine the numerators over the common denominator.

$3 t = \frac{- 8 + 13 \cdot 3}{3}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $13$ by $3$ .

$3 t = \frac{- 8 + 39}{3}$

Step 4.5.2

Add $- 8$ and $39$ .

$3 t = \frac{31}{3}$

Step 5

Divide each term in $3 t = \frac{31}{3}$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 t = \frac{31}{3}$ by $3$ .

$\frac{3 t}{3} = \frac{\frac{31}{3}}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 t}{3} = \frac{\frac{31}{3}}{3}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\frac{31}{3}}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$t = \frac{31}{3} \cdot \frac{1}{3}$

Step 5.3.2

Multiply $\frac{31}{3} \cdot \frac{1}{3}$ .

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Step 5.3.2.1

Multiply $\frac{31}{3}$ by $\frac{1}{3}$ .

$t = \frac{31}{3 \cdot 3}$

Step 5.3.2.2

Multiply $3$ by $3$ .

$t = \frac{31}{9}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{31}{9}$

Decimal Form:

$t = 3.\overline{4}$

Mixed Number Form:

$t = 3 \frac{4}{9}$