Finite Math Examples

$- \frac{3}{4} - \frac{- 8 x + 21}{12} \cdot e^{- \frac{2}{3} t} = 0$

Step 1

Simplify $- \frac{3}{4} - \frac{- 8 x + 21}{12} \cdot e^{- \frac{2}{3} t}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$- \frac{3}{4} - \frac{- 8 x + 21}{12} \cdot e^{- \frac{t \cdot 2}{3}} = 0$

Step 1.1.2

Move $2$ to the left of $t$ .

$- \frac{3}{4} - \frac{- 8 x + 21}{12} \cdot e^{- \frac{2 \cdot t}{3}} = 0$

Step 1.1.3

Combine $e^{- \frac{2 t}{3}}$ and $\frac{- 8 x + 21}{12}$ .

$- \frac{3}{4} - \frac{e^{- \frac{2 t}{3}} (- 8 x + 21)}{12} = 0$

Step 1.2

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

$- \frac{3}{4} \cdot \frac{3}{3} - \frac{e^{- \frac{2 t}{3}} (- 8 x + 21)}{12} = 0$

Step 1.3

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

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Step 1.3.1

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

$- \frac{3 \cdot 3}{4 \cdot 3} - \frac{e^{- \frac{2 t}{3}} (- 8 x + 21)}{12} = 0$

Step 1.3.2

Multiply $4$ by $3$ .

$- \frac{3 \cdot 3}{12} - \frac{e^{- \frac{2 t}{3}} (- 8 x + 21)}{12} = 0$

Step 1.4

Combine the numerators over the common denominator.

$\frac{- 3 \cdot 3 - e^{- \frac{2 t}{3}} (- 8 x + 21)}{12} = 0$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 3$ by $3$ .

$\frac{- 9 - e^{- \frac{2 t}{3}} (- 8 x + 21)}{12} = 0$

Step 1.5.2

Apply the distributive property.

$\frac{- 9 - e^{- \frac{2 t}{3}} (- 8 x) - e^{- \frac{2 t}{3}} \cdot 21}{12} = 0$

Step 1.5.3

Rewrite using the commutative property of multiplication.

$\frac{- 9 - 1 \cdot - 8 e^{- \frac{2 t}{3}} x - e^{- \frac{2 t}{3}} \cdot 21}{12} = 0$

Step 1.5.4

Multiply $21$ by $- 1$ .

$\frac{- 9 - 1 \cdot - 8 e^{- \frac{2 t}{3}} x - 21 e^{- \frac{2 t}{3}}}{12} = 0$

Step 1.5.5

Multiply $- 1$ by $- 8$ .

$\frac{- 9 + 8 e^{- \frac{2 t}{3}} x - 21 e^{- \frac{2 t}{3}}}{12} = 0$

Step 1.6

Simplify with factoring out.

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Step 1.6.1

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{- 1 (9) + 8 e^{- \frac{2 t}{3}} x - 21 e^{- \frac{2 t}{3}}}{12} = 0$

Step 1.6.2

Factor $- 1$ out of $8 e^{- \frac{2 t}{3}} x$ .

$\frac{- 1 (9) - (- 8 e^{- \frac{2 t}{3}} x) - 21 e^{- \frac{2 t}{3}}}{12} = 0$

Step 1.6.3

Factor $- 1$ out of $- 1 (9) - (- 8 e^{- \frac{2 t}{3}} x)$ .

$\frac{- 1 (9 - 8 e^{- \frac{2 t}{3}} x) - 21 e^{- \frac{2 t}{3}}}{12} = 0$

Step 1.6.4

Factor $- 1$ out of $- 21 e^{- \frac{2 t}{3}}$ .

$\frac{- 1 (9 - 8 e^{- \frac{2 t}{3}} x) - (21 e^{- \frac{2 t}{3}})}{12} = 0$

Step 1.6.5

Factor $- 1$ out of $- 1 (9 - 8 e^{- \frac{2 t}{3}} x) - (21 e^{- \frac{2 t}{3}})$ .

$\frac{- 1 (9 - 8 e^{- \frac{2 t}{3}} x + 21 e^{- \frac{2 t}{3}})}{12} = 0$

Step 1.6.6

Simplify the expression.

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Step 1.6.6.1

Move the negative in front of the fraction.

$- \frac{9 - 8 e^{- \frac{2 t}{3}} x + 21 e^{- \frac{2 t}{3}}}{12} = 0$

Step 1.6.6.2

Reorder factors in $- \frac{9 - 8 e^{- \frac{2 t}{3}} x + 21 e^{- \frac{2 t}{3}}}{12}$ .

$- \frac{9 - 8 x e^{- \frac{2 t}{3}} + 21 e^{- \frac{2 t}{3}}}{12} = 0$

Step 2

Set the numerator equal to zero.

$9 - 8 x e^{- \frac{2 t}{3}} + 21 e^{- \frac{2 t}{3}} = 0$

Step 3

Solve the equation for $t$ .

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Step 3.1

Subtract $9$ from both sides of the equation.

$- 8 x e^{- \frac{2 t}{3}} + 21 e^{- \frac{2 t}{3}} = - 9$

Step 3.2

Factor $e^{- \frac{2 t}{3}}$ out of $- 8 x e^{- \frac{2 t}{3}} + 21 e^{- \frac{2 t}{3}}$ .

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Step 3.2.1

Factor $e^{- \frac{2 t}{3}}$ out of $- 8 x e^{- \frac{2 t}{3}}$ .

$e^{- \frac{2 t}{3}} (- 8 x) + 21 e^{- \frac{2 t}{3}} = - 9$

Step 3.2.2

Factor $e^{- \frac{2 t}{3}}$ out of $21 e^{- \frac{2 t}{3}}$ .

$e^{- \frac{2 t}{3}} (- 8 x) + e^{- \frac{2 t}{3}} \cdot 21 = - 9$

Step 3.2.3

Factor $e^{- \frac{2 t}{3}}$ out of $e^{- \frac{2 t}{3}} (- 8 x) + e^{- \frac{2 t}{3}} \cdot 21$ .

$e^{- \frac{2 t}{3}} (- 8 x + 21) = - 9$

Step 3.3

Divide each term in $e^{- \frac{2 t}{3}} (- 8 x + 21) = - 9$ by $- 8 x + 21$ and simplify.

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Step 3.3.1

Divide each term in $e^{- \frac{2 t}{3}} (- 8 x + 21) = - 9$ by $- 8 x + 21$ .

$\frac{e^{- \frac{2 t}{3}} (- 8 x + 21)}{- 8 x + 21} = \frac{- 9}{- 8 x + 21}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 8 x + 21$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{e^{- \frac{2 t}{3}} (- 8 x + 21)}{- 8 x + 21} = \frac{- 9}{- 8 x + 21}$

Step 3.3.2.1.2

Divide $e^{- \frac{2 t}{3}}$ by $1$ .

$e^{- \frac{2 t}{3}} = \frac{- 9}{- 8 x + 21}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$e^{- \frac{2 t}{3}} = - \frac{9}{- 8 x + 21}$

Step 3.3.3.2

Factor $- 1$ out of $- 8 x$ .

$e^{- \frac{2 t}{3}} = - \frac{9}{- (8 x) + 21}$

Step 3.3.3.3

Rewrite $21$ as $- 1 (- 21)$ .

$e^{- \frac{2 t}{3}} = - \frac{9}{- (8 x) - 1 (- 21)}$

Step 3.3.3.4

Factor $- 1$ out of $- (8 x) - 1 (- 21)$ .

$e^{- \frac{2 t}{3}} = - \frac{9}{- (8 x - 21)}$

Step 3.3.3.5

Simplify the expression.

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Step 3.3.3.5.1

Rewrite $- (8 x - 21)$ as $- 1 (8 x - 21)$ .

$e^{- \frac{2 t}{3}} = - \frac{9}{- 1 (8 x - 21)}$

Step 3.3.3.5.2

Move the negative in front of the fraction.

$e^{- \frac{2 t}{3}} = - - \frac{9}{8 x - 21}$

Step 3.3.3.5.3

Multiply $- 1$ by $- 1$ .

$e^{- \frac{2 t}{3}} = 1 \frac{9}{8 x - 21}$

Step 3.3.3.5.4

Multiply $\frac{9}{8 x - 21}$ by $1$ .

$e^{- \frac{2 t}{3}} = \frac{9}{8 x - 21}$

Step 3.4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- \frac{2 t}{3}}) = \ln (\frac{9}{8 x - 21})$

Step 3.5

Expand the left side.

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Step 3.5.1

Expand $\ln (e^{- \frac{2 t}{3}})$ by moving $- \frac{2 t}{3}$ outside the logarithm.

$- \frac{2 t}{3} \ln (e) = \ln (\frac{9}{8 x - 21})$

Step 3.5.2

The natural logarithm of $e$ is $1$ .

$- \frac{2 t}{3} \cdot 1 = \ln (\frac{9}{8 x - 21})$

Step 3.5.3

Multiply $- 1$ by $1$ .

$- \frac{2 t}{3} = \ln (\frac{9}{8 x - 21})$

Step 3.6

Multiply both sides of the equation by $- \frac{3}{2}$ .

$- \frac{3}{2} (- \frac{2 t}{3}) = - \frac{3}{2} \ln (\frac{9}{8 x - 21})$

Step 3.7

Simplify both sides of the equation.

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Step 3.7.1

Simplify the left side.

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Step 3.7.1.1

Simplify $- \frac{3}{2} (- \frac{2 t}{3})$ .

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Step 3.7.1.1.1

Cancel the common factor of $3$ .

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Step 3.7.1.1.1.1

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

$\frac{- 3}{2} (- \frac{2 t}{3}) = - \frac{3}{2} \ln (\frac{9}{8 x - 21})$

Step 3.7.1.1.1.2

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

$\frac{- 3}{2} \cdot \frac{- 2 t}{3} = - \frac{3}{2} \ln (\frac{9}{8 x - 21})$

Step 3.7.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 t}{3} = - \frac{3}{2} \ln (\frac{9}{8 x - 21})$

Step 3.7.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 t}{3} = - \frac{3}{2} \ln (\frac{9}{8 x - 21})$

Step 3.7.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2} (- 2 t) = - \frac{3}{2} \ln (\frac{9}{8 x - 21})$

Step 3.7.1.1.2

Cancel the common factor of $2$ .

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Step 3.7.1.1.2.1

Factor $2$ out of $- 2 t$ .

$\frac{- 1}{2} (2 (- t)) = - \frac{3}{2} \ln (\frac{9}{8 x - 21})$

Step 3.7.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2} (2 (- t)) = - \frac{3}{2} \ln (\frac{9}{8 x - 21})$

Step 3.7.1.1.2.3

Rewrite the expression.

$- - t = - \frac{3}{2} \ln (\frac{9}{8 x - 21})$

Step 3.7.1.1.3

Multiply.

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Step 3.7.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 t = - \frac{3}{2} \ln (\frac{9}{8 x - 21})$

Step 3.7.1.1.3.2

Multiply $t$ by $1$ .

$t = - \frac{3}{2} \ln (\frac{9}{8 x - 21})$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Simplify $- \frac{3}{2} \ln (\frac{9}{8 x - 21})$ .

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Step 3.7.2.1.1

Combine $\ln (\frac{9}{8 x - 21})$ and $\frac{3}{2}$ .

$t = - \frac{\ln (\frac{9}{8 x - 21}) \cdot 3}{2}$

Step 3.7.2.1.2

Move $3$ to the left of $\ln (\frac{9}{8 x - 21})$ .

$t = - \frac{3 \ln (\frac{9}{8 x - 21})}{2}$