Solve {l}{1081.14/(1+x)^5+1081.14/(1+x)^6=0}{text{Solvefor}ytext{where}}{y=-6299+1081.14/1+x+1081.14/(1+x)^2+frac{1081.14}{(1+x)^3}+frac{1081*14}{{(1+x)}^4}}

$$\left. \begin{array} { l } { \frac{1081.14}{{(1 + x)} ^ {5}} + \frac{1081.14}{{(1 + x)} ^ {6}} = 0 }\\ { \text{Solve for } y \text{ where} } \\ { y = -6299 + \frac{1081.14}{1 + x} + \frac{1081.14}{{(1 + x)} ^ {2}} + \frac{1081.14}{{(1 + x)} ^ {3}} + \frac{1081 \cdot 14}{{(1 + x)} ^ {4}} } \end{array} \right.$$

Solve for x, y

$$y=7753.86$$

Solution Steps

Consider the first equation. Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)^{6}, the least common multiple of \left(1+x\right)^{5},\left(1+x\right)^{6}.

$$\left(x+1\right)\times 1081.14+1081.14=0$$

Use the distributive property to multiply x+1 by 1081.14.

$$1081.14x+1081.14+1081.14=0$$

Add 1081.14 and 1081.14 to get 2162.28.

$$1081.14x+2162.28=0$$

Subtract 2162.28 from both sides. Anything subtracted from zero gives its negation.

$$1081.14x=-2162.28$$

Divide both sides by 1081.14.

$$x=\frac{-2162.28}{1081.14}$$

Expand \frac{-2162.28}{1081.14} by multiplying both numerator and the denominator by 100.

$$x=\frac{-216228}{108114}$$

Divide -216228 by 108114 to get -2.

$$x=-2$$

Consider the second equation. Insert the known values of variables into the equation.

$$y=-6299+\frac{1081.14}{1-2}+\frac{1081.14}{\left(1-2\right)^{2}}+\frac{1081.14}{\left(1-2\right)^{3}}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Subtract 2 from 1 to get -1.

$$y=-6299+\frac{1081.14}{-1}+\frac{1081.14}{\left(1-2\right)^{2}}+\frac{1081.14}{\left(1-2\right)^{3}}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Expand \frac{1081.14}{-1} by multiplying both numerator and the denominator by 100.

$$y=-6299+\frac{108114}{-100}+\frac{1081.14}{\left(1-2\right)^{2}}+\frac{1081.14}{\left(1-2\right)^{3}}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Reduce the fraction \frac{108114}{-100} to lowest terms by extracting and canceling out 2.

$$y=-6299-\frac{54057}{50}+\frac{1081.14}{\left(1-2\right)^{2}}+\frac{1081.14}{\left(1-2\right)^{3}}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Subtract \frac{54057}{50} from -6299 to get -\frac{369007}{50}.

$$y=-\frac{369007}{50}+\frac{1081.14}{\left(1-2\right)^{2}}+\frac{1081.14}{\left(1-2\right)^{3}}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Subtract 2 from 1 to get -1.

$$y=-\frac{369007}{50}+\frac{1081.14}{\left(-1\right)^{2}}+\frac{1081.14}{\left(1-2\right)^{3}}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Calculate -1 to the power of 2 and get 1.

$$y=-\frac{369007}{50}+\frac{1081.14}{1}+\frac{1081.14}{\left(1-2\right)^{3}}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Anything divided by one gives itself.

$$y=-\frac{369007}{50}+1081.14+\frac{1081.14}{\left(1-2\right)^{3}}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Add -\frac{369007}{50} and 1081.14 to get -6299.

$$y=-6299+\frac{1081.14}{\left(1-2\right)^{3}}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Subtract 2 from 1 to get -1.

$$y=-6299+\frac{1081.14}{\left(-1\right)^{3}}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Calculate -1 to the power of 3 and get -1.

$$y=-6299+\frac{1081.14}{-1}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Expand \frac{1081.14}{-1} by multiplying both numerator and the denominator by 100.

$$y=-6299+\frac{108114}{-100}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Reduce the fraction \frac{108114}{-100} to lowest terms by extracting and canceling out 2.

$$y=-6299-\frac{54057}{50}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Subtract \frac{54057}{50} from -6299 to get -\frac{369007}{50}.

$$y=-\frac{369007}{50}+\frac{1081\times 14}{\left(1-2\right)^{4}}$$

Multiply 1081 and 14 to get 15134.

$$y=-\frac{369007}{50}+\frac{15134}{\left(1-2\right)^{4}}$$

Subtract 2 from 1 to get -1.

$$y=-\frac{369007}{50}+\frac{15134}{\left(-1\right)^{4}}$$

Calculate -1 to the power of 4 and get 1.

$$y=-\frac{369007}{50}+\frac{15134}{1}$$

Anything divided by one gives itself.

$$y=-\frac{369007}{50}+15134$$

Add -\frac{369007}{50} and 15134 to get \frac{387693}{50}.

$$y=\frac{387693}{50}$$

The system is now solved.

$$x=-2 y=\frac{387693}{50}$$