Precalculus Examples

$\frac{1}{t} + \frac{1}{3 t} + \frac{1}{5 t} = 9$

Step 1

Subtract $9$ from both sides of the equation.

$\frac{1}{t} + \frac{1}{3 t} + \frac{1}{5 t} - 9 = 0$

Step 2

Simplify $\frac{1}{t} + \frac{1}{3 t} + \frac{1}{5 t} - 9$ .

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Step 2.1

To write $\frac{1}{t}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{t} \cdot \frac{3}{3} + \frac{1}{3 t} + \frac{1}{5 t} - 9 = 0$

Step 2.2

Write each expression with a common denominator of $t \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1

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

$\frac{3}{t \cdot 3} + \frac{1}{3 t} + \frac{1}{5 t} - 9 = 0$

Step 2.2.2

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

$\frac{3}{3 t} + \frac{1}{3 t} + \frac{1}{5 t} - 9 = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{3 + 1}{3 t} + \frac{1}{5 t} - 9 = 0$

Step 2.4

Add $3$ and $1$ .

$\frac{4}{3 t} + \frac{1}{5 t} - 9 = 0$

Step 2.5

To write $\frac{4}{3 t}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4}{3 t} \cdot \frac{5}{5} + \frac{1}{5 t} - 9 = 0$

Step 2.6

To write $\frac{1}{5 t}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4}{3 t} \cdot \frac{5}{5} + \frac{1}{5 t} \cdot \frac{3}{3} - 9 = 0$

Step 2.7

Write each expression with a common denominator of $15 t$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.7.1

Multiply $\frac{4}{3 t}$ by $\frac{5}{5}$ .

$\frac{4 \cdot 5}{3 t \cdot 5} + \frac{1}{5 t} \cdot \frac{3}{3} - 9 = 0$

Step 2.7.2

Multiply $5$ by $3$ .

$\frac{4 \cdot 5}{15 t} + \frac{1}{5 t} \cdot \frac{3}{3} - 9 = 0$

Step 2.7.3

Multiply $\frac{1}{5 t}$ by $\frac{3}{3}$ .

$\frac{4 \cdot 5}{15 t} + \frac{3}{5 t \cdot 3} - 9 = 0$

Step 2.7.4

Multiply $3$ by $5$ .

$\frac{4 \cdot 5}{15 t} + \frac{3}{15 t} - 9 = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{4 \cdot 5 + 3}{15 t} - 9 = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $4$ by $5$ .

$\frac{20 + 3}{15 t} - 9 = 0$

Step 2.9.2

Add $20$ and $3$ .

$\frac{23}{15 t} - 9 = 0$

Step 3

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{15 t}{15 t}$ .

$\frac{23}{15 t} - 9 \cdot \frac{15 t}{15 t} = 0$

Step 4

Combine $- 9$ and $\frac{15 t}{15 t}$ .

$\frac{23}{15 t} + \frac{- 9 (15 t)}{15 t} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{23 - 9 (15 t)}{15 t} = 0$

Step 6

Remove parentheses.

$\frac{23 - 9 \cdot (15 t)}{15 t} = 0$

Step 7

Multiply $- 9$ by $15$ .

$\frac{23 - 135 t}{15 t} = 0$

Step 8

Set the numerator equal to zero.

$23 - 135 t = 0$

Step 9

Solve the equation for $t$ .

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Step 9.1

Subtract $23$ from both sides of the equation.

$- 135 t = - 23$

Step 9.2

Divide each term in $- 135 t = - 23$ by $- 135$ and simplify.

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Step 9.2.1

Divide each term in $- 135 t = - 23$ by $- 135$ .

$\frac{- 135 t}{- 135} = \frac{- 23}{- 135}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $- 135$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{- 135 t}{- 135} = \frac{- 23}{- 135}$

Step 9.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 23}{- 135}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Dividing two negative values results in a positive value.

$t = \frac{23}{135}$