Finite Math Examples

$y = \frac{2425.01 (1 - {(1 + 0.11)}^{- (12 - x)})}{\frac{0.11}{2}}$

Step 1

Rewrite the equation as $\frac{2425.01 (1 - {(1 + 0.11)}^{- (12 - x)})}{\frac{0.11}{2}} = y$ .

$\frac{2425.01 (1 - {(1 + 0.11)}^{- (12 - x)})}{\frac{0.11}{2}} = y$

Step 2

Multiply both sides by $\frac{0.11}{2}$ .

$\frac{2425.01 (1 - {(1 + 0.11)}^{- (12 - x)})}{\frac{0.11}{2}} \cdot \frac{0.11}{2} = y \frac{0.11}{2}$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{2425.01 (1 - {(1 + 0.11)}^{- (12 - x)})}{\frac{0.11}{2}} \cdot \frac{0.11}{2}$ .

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Step 3.1.1.1

Combine fractions.

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Step 3.1.1.1.1

Combine.

$\frac{2425.01 (1 - {(1 + 0.11)}^{- (12 - x)}) \cdot 0.11}{\frac{0.11}{2} \cdot 2} = y \frac{0.11}{2}$

Step 3.1.1.1.2

Multiply $0.11$ by $2425.01$ .

$\frac{266.7511 (1 - {(1 + 0.11)}^{- (12 - x)})}{\frac{0.11}{2} \cdot 2} = y \frac{0.11}{2}$

Step 3.1.1.1.3

Combine $\frac{0.11}{2}$ and $2$ .

$\frac{266.7511 (1 - {(1 + 0.11)}^{- (12 - x)})}{\frac{0.11 \cdot 2}{2}} = y \frac{0.11}{2}$

Step 3.1.1.1.4

Multiply $0.11$ by $2$ .

$\frac{266.7511 (1 - {(1 + 0.11)}^{- (12 - x)})}{\frac{0.22}{2}} = y \frac{0.11}{2}$

Step 3.1.1.2

Simplify the numerator.

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Step 3.1.1.2.1

Add $1$ and $0.11$ .

$\frac{266.7511 (1 - {1.11}^{- (12 - x)})}{\frac{0.22}{2}} = y \frac{0.11}{2}$

Step 3.1.1.2.2

Apply the distributive property.

$\frac{266.7511 (1 - {1.11}^{- 1 \cdot 12 - - x})}{\frac{0.22}{2}} = y \frac{0.11}{2}$

Step 3.1.1.2.3

Multiply $- 1$ by $12$ .

$\frac{266.7511 (1 - {1.11}^{- 12 - - x})}{\frac{0.22}{2}} = y \frac{0.11}{2}$

Step 3.1.1.2.4

Multiply $- - x$ .

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Step 3.1.1.2.4.1

Multiply $- 1$ by $- 1$ .

$\frac{266.7511 (1 - {1.11}^{- 12 + 1 x})}{\frac{0.22}{2}} = y \frac{0.11}{2}$

Step 3.1.1.2.4.2

Multiply $x$ by $1$ .

$\frac{266.7511 (1 - {1.11}^{- 12 + x})}{\frac{0.22}{2}} = y \frac{0.11}{2}$

Step 3.1.1.3

Simplify with factoring out.

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Step 3.1.1.3.1

Divide $0.22$ by $2$ .

$\frac{266.7511 (1 - {1.11}^{- 12 + x})}{0.11} = y \frac{0.11}{2}$

Step 3.1.1.3.2

Factor $0.11$ out of $0.11$ .

$\frac{266.7511 (1 - {1.11}^{- 12 + x})}{0.11 (1)} = y \frac{0.11}{2}$

Step 3.1.1.4

Separate fractions.

$\frac{266.7511}{0.11} \cdot \frac{1 - {1.11}^{- 12 + x}}{1} = y \frac{0.11}{2}$

Step 3.1.1.5

Simplify the expression.

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Step 3.1.1.5.1

Divide $266.7511$ by $0.11$ .

$2425.01 \frac{1 - {1.11}^{- 12 + x}}{1} = y \frac{0.11}{2}$

Step 3.1.1.5.2

Divide $1 - {1.11}^{- 12 + x}$ by $1$ .

$2425.01 (1 - {1.11}^{- 12 + x}) = y \frac{0.11}{2}$

Step 3.1.1.6

Apply the distributive property.

$2425.01 \cdot 1 + 2425.01 (- {1.11}^{- 12 + x}) = y \frac{0.11}{2}$

Step 3.1.1.7

Simplify the expression.

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Step 3.1.1.7.1

Multiply $2425.01$ by $1$ .

$2425.01 + 2425.01 (- {1.11}^{- 12 + x}) = y \frac{0.11}{2}$

Step 3.1.1.7.2

Multiply $- 1$ by $2425.01$ .

$2425.01 - 2425.01 \cdot {1.11}^{- 12 + x} = y \frac{0.11}{2}$

Step 3.1.1.7.3

Reorder $- 12$ and $x$ .

$2425.01 - 2425.01 \cdot {1.11}^{x - 12} = y \frac{0.11}{2}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $y \frac{0.11}{2}$ .

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Step 3.2.1.1

Divide $0.11$ by $2$ .

$2425.01 - 2425.01 \cdot {1.11}^{x - 12} = y \cdot 0.055$

Step 3.2.1.2

Move $0.055$ to the left of $y$ .

$2425.01 - 2425.01 \cdot {1.11}^{x - 12} = 0.055 y$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $2425.01$ from both sides of the equation.

$- 2425.01 \cdot {1.11}^{x - 12} = 0.055 y - 2425.01$

Step 4.2

Divide each term in $- 2425.01 \cdot {1.11}^{x - 12} = 0.055 y - 2425.01$ by $- 2425.01$ and simplify.

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Step 4.2.1

Divide each term in $- 2425.01 \cdot {1.11}^{x - 12} = 0.055 y - 2425.01$ by $- 2425.01$ .

$\frac{- 2425.01 \cdot {1.11}^{x - 12}}{- 2425.01} = \frac{0.055 y}{- 2425.01} + \frac{- 2425.01}{- 2425.01}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 2425.01$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 2425.01 \cdot {1.11}^{x - 12}}{- 2425.01} = \frac{0.055 y}{- 2425.01} + \frac{- 2425.01}{- 2425.01}$

Step 4.2.2.1.2

Divide ${1.11}^{x - 12}$ by $1$ .

${1.11}^{x - 12} = \frac{0.055 y}{- 2425.01} + \frac{- 2425.01}{- 2425.01}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Move the negative in front of the fraction.

${1.11}^{x - 12} = - \frac{0.055 y}{2425.01} + \frac{- 2425.01}{- 2425.01}$

Step 4.2.3.1.2

Factor $0.055$ out of $0.055 y$ .

${1.11}^{x - 12} = - \frac{0.055 (y)}{2425.01} + \frac{- 2425.01}{- 2425.01}$

Step 4.2.3.1.3

Factor $2425.01$ out of $2425.01$ .

${1.11}^{x - 12} = - \frac{0.055 (y)}{2425.01 (1)} + \frac{- 2425.01}{- 2425.01}$

Step 4.2.3.1.4

Separate fractions.

${1.11}^{x - 12} = - (\frac{0.055}{2425.01} \cdot \frac{y}{1}) + \frac{- 2425.01}{- 2425.01}$

Step 4.2.3.1.5

Divide $0.055$ by $2425.01$ .

${1.11}^{x - 12} = - (0.00002268 \frac{y}{1}) + \frac{- 2425.01}{- 2425.01}$

Step 4.2.3.1.6

Divide $y$ by $1$ .

${1.11}^{x - 12} = - (0.00002268 y) + \frac{- 2425.01}{- 2425.01}$

Step 4.2.3.1.7

Multiply $0.00002268$ by $- 1$ .

${1.11}^{x - 12} = - 0.00002268 y + \frac{- 2425.01}{- 2425.01}$

Step 4.2.3.1.8

Divide $- 2425.01$ by $- 2425.01$ .

${1.11}^{x - 12} = - 0.00002268 y + 1$

Step 4.3

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

$\ln ({1.11}^{x - 12}) = \ln (- 0.00002268 y + 1)$

Step 4.4

Expand $\ln ({1.11}^{x - 12})$ by moving $x - 12$ outside the logarithm.

$(x - 12) \ln (1.11) = \ln (- 0.00002268 y + 1)$

Step 4.5

Simplify the left side.

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Step 4.5.1

Apply the distributive property.

$x \ln (1.11) - 12 \ln (1.11) = \ln (- 0.00002268 y + 1)$

Step 4.6

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (1.11) - 12 \ln (1.11) - \ln (- 0.00002268 y + 1) = 0$

Step 4.7

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

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Step 4.7.1

Add $12 \ln (1.11)$ to both sides of the equation.

$x \ln (1.11) - \ln (- 0.00002268 y + 1) = 12 \ln (1.11)$

Step 4.7.2

Add $\ln (- 0.00002268 y + 1)$ to both sides of the equation.

$x \ln (1.11) = 12 \ln (1.11) + \ln (- 0.00002268 y + 1)$

Step 4.8

Divide each term in $x \ln (1.11) = 12 \ln (1.11) + \ln (- 0.00002268 y + 1)$ by $\ln (1.11)$ and simplify.

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Step 4.8.1

Divide each term in $x \ln (1.11) = 12 \ln (1.11) + \ln (- 0.00002268 y + 1)$ by $\ln (1.11)$ .

$\frac{x \ln (1.11)}{\ln (1.11)} = \frac{12 \ln (1.11)}{\ln (1.11)} + \frac{\ln (- 0.00002268 y + 1)}{\ln (1.11)}$

Step 4.8.2

Simplify the left side.

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Step 4.8.2.1

Cancel the common factor of $\ln (1.11)$ .

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Step 4.8.2.1.1

Cancel the common factor.

$\frac{x \ln (1.11)}{\ln (1.11)} = \frac{12 \ln (1.11)}{\ln (1.11)} + \frac{\ln (- 0.00002268 y + 1)}{\ln (1.11)}$

Step 4.8.2.1.2

Divide $x$ by $1$ .

$x = \frac{12 \ln (1.11)}{\ln (1.11)} + \frac{\ln (- 0.00002268 y + 1)}{\ln (1.11)}$

Step 4.8.3

Simplify the right side.

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Step 4.8.3.1

Cancel the common factor of $\ln (1.11)$ .

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Step 4.8.3.1.1

Cancel the common factor.

$x = \frac{12 \ln (1.11)}{\ln (1.11)} + \frac{\ln (- 0.00002268 y + 1)}{\ln (1.11)}$

Step 4.8.3.1.2

Divide $12$ by $1$ .

$x = 12 + \frac{\ln (- 0.00002268 y + 1)}{\ln (1.11)}$