Precalculus Examples

$\frac{1}{8} \cdot (4 t - 5) = \frac{11}{8} t - \frac{t + 5}{16}$

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.

$\frac{11}{8} t - \frac{t + 5}{16} = \frac{1}{8} \cdot (4 t - 5)$

Step 2

Simplify $\frac{11}{8} t - \frac{t + 5}{16}$ .

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Step 2.1

Combine $\frac{11}{8}$ and $t$ .

$\frac{11 t}{8} - \frac{t + 5}{16} = \frac{1}{8} \cdot (4 t - 5)$

Step 2.2

To write $\frac{11 t}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{11 t}{8} \cdot \frac{2}{2} - \frac{t + 5}{16} = \frac{1}{8} \cdot (4 t - 5)$

Step 2.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{11 t}{8}$ by $\frac{2}{2}$ .

$\frac{11 t \cdot 2}{8 \cdot 2} - \frac{t + 5}{16} = \frac{1}{8} \cdot (4 t - 5)$

Step 2.3.2

Multiply $8$ by $2$ .

$\frac{11 t \cdot 2}{16} - \frac{t + 5}{16} = \frac{1}{8} \cdot (4 t - 5)$

Step 2.4

Combine the numerators over the common denominator.

$\frac{11 t \cdot 2 - (t + 5)}{16} = \frac{1}{8} \cdot (4 t - 5)$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $2$ by $11$ .

$\frac{22 t - (t + 5)}{16} = \frac{1}{8} \cdot (4 t - 5)$

Step 2.5.2

Apply the distributive property.

$\frac{22 t - t - 1 \cdot 5}{16} = \frac{1}{8} \cdot (4 t - 5)$

Step 2.5.3

Multiply $- 1$ by $5$ .

$\frac{22 t - t - 5}{16} = \frac{1}{8} \cdot (4 t - 5)$

Step 2.5.4

Subtract $t$ from $22 t$ .

$\frac{21 t - 5}{16} = \frac{1}{8} \cdot (4 t - 5)$

Step 3

Simplify $\frac{1}{8} \cdot (4 t - 5)$ .

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Step 3.1

Apply the distributive property.

$\frac{21 t - 5}{16} = \frac{1}{8} (4 t) + \frac{1}{8} \cdot - 5$

Step 3.2

Cancel the common factor of $4$ .

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Step 3.2.1

Factor $4$ out of $8$ .

$\frac{21 t - 5}{16} = \frac{1}{4 (2)} (4 t) + \frac{1}{8} \cdot - 5$

Step 3.2.2

Factor $4$ out of $4 t$ .

$\frac{21 t - 5}{16} = \frac{1}{4 (2)} (4 (t)) + \frac{1}{8} \cdot - 5$

Step 3.2.3

Cancel the common factor.

$\frac{21 t - 5}{16} = \frac{1}{4 \cdot 2} (4 t) + \frac{1}{8} \cdot - 5$

Step 3.2.4

Rewrite the expression.

$\frac{21 t - 5}{16} = \frac{1}{2} t + \frac{1}{8} \cdot - 5$

Step 3.3

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

$\frac{21 t - 5}{16} = \frac{t}{2} + \frac{1}{8} \cdot - 5$

Step 3.4

Combine $\frac{1}{8}$ and $- 5$ .

$\frac{21 t - 5}{16} = \frac{t}{2} + \frac{- 5}{8}$

Step 3.5

Move the negative in front of the fraction.

$\frac{21 t - 5}{16} = \frac{t}{2} - \frac{5}{8}$

Step 4

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

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Step 4.1

Subtract $\frac{t}{2}$ from both sides of the equation.

$\frac{21 t - 5}{16} - \frac{t}{2} = - \frac{5}{8}$

Step 4.2

To write $- \frac{t}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{21 t - 5}{16} - \frac{t}{2} \cdot \frac{8}{8} = - \frac{5}{8}$

Step 4.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{t}{2}$ by $\frac{8}{8}$ .

$\frac{21 t - 5}{16} - \frac{t \cdot 8}{2 \cdot 8} = - \frac{5}{8}$

Step 4.3.2

Multiply $2$ by $8$ .

$\frac{21 t - 5}{16} - \frac{t \cdot 8}{16} = - \frac{5}{8}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{21 t - 5 - t \cdot 8}{16} = - \frac{5}{8}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $8$ by $- 1$ .

$\frac{21 t - 5 - 8 t}{16} = - \frac{5}{8}$

Step 4.5.2

Subtract $8 t$ from $21 t$ .

$\frac{13 t - 5}{16} = - \frac{5}{8}$

Step 5

Multiply both sides by $16$ .

$\frac{13 t - 5}{16} \cdot 16 = - \frac{5}{8} \cdot 16$

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $16$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{13 t - 5}{16} \cdot 16 = - \frac{5}{8} \cdot 16$

Step 6.1.1.2

Rewrite the expression.

$13 t - 5 = - \frac{5}{8} \cdot 16$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{5}{8} \cdot 16$ .

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Step 6.2.1.1

Cancel the common factor of $8$ .

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Step 6.2.1.1.1

Move the leading negative in $- \frac{5}{8}$ into the numerator.

$13 t - 5 = \frac{- 5}{8} \cdot 16$

Step 6.2.1.1.2

Factor $8$ out of $16$ .

$13 t - 5 = \frac{- 5}{8} \cdot (8 (2))$

Step 6.2.1.1.3

Cancel the common factor.

$13 t - 5 = \frac{- 5}{8} \cdot (8 \cdot 2)$

Step 6.2.1.1.4

Rewrite the expression.

$13 t - 5 = - 5 \cdot 2$

Step 6.2.1.2

Multiply $- 5$ by $2$ .

$13 t - 5 = - 10$

Step 7

Solve for $t$ .

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Step 7.1

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

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Step 7.1.1

Add $5$ to both sides of the equation.

$13 t = - 10 + 5$

Step 7.1.2

Add $- 10$ and $5$ .

$13 t = - 5$

Step 7.2

Divide each term in $13 t = - 5$ by $13$ and simplify.

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Step 7.2.1

Divide each term in $13 t = - 5$ by $13$ .

$\frac{13 t}{13} = \frac{- 5}{13}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $13$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{13 t}{13} = \frac{- 5}{13}$

Step 7.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 5}{13}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Move the negative in front of the fraction.

$t = - \frac{5}{13}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$t = - \frac{5}{13}$

Decimal Form:

$t = - 0.\overline{384615}$