Basic Math Examples

1.1 = v sin (45) (\frac{4.4}{v cos (45)}) - 4.9 {(\frac{4.4}{v cos (45)})}^{2}

$1.1 = v \sin (45) (\frac{4.4}{v \cos (45)}) - 4.9 {(\frac{4.4}{v \cos (45)})}^{2}$

Step 1

Rewrite the equation as

$v \sin (45) (\frac{4.4}{v \cos (45)}) - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} = 1.1$ .

$v \sin (45) (\frac{4.4}{v \cos (45)}) - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} = 1.1$

Step 2

Subtract

$1.1$ from both sides of the equation.

$v \sin (45) (\frac{4.4}{v \cos (45)}) - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3

Simplify the left side of the equation.

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Step 3.1

Simplify each term.

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Step 3.1.1

The exact value of

$\cos (45)$ is

$\frac{\sqrt{2}}{2}$ .

$v \sin (45) (\frac{4.4}{v (\frac{\sqrt{2}}{2})}) - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.2

Combine

$v$ and

$\frac{\sqrt{2}}{2}$ .

$v \sin (45) (\frac{4.4}{\frac{v \sqrt{2}}{2}}) - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.3

The exact value of

$\sin (45)$ is

$\frac{\sqrt{2}}{2}$ .

$v (\frac{\sqrt{2}}{2}) \cdot \frac{4.4}{\frac{v \sqrt{2}}{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.4

Combine

$v$ and

$\frac{\sqrt{2}}{2}$ .

$\frac{v \sqrt{2}}{2} \cdot \frac{4.4}{\frac{v \sqrt{2}}{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{v \sqrt{2}}{2} \cdot (4.4 (\frac{2}{v \sqrt{2}})) - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.6

Rewrite using the commutative property of multiplication.

$4.4 (\frac{v \sqrt{2}}{2}) \cdot \frac{2}{v \sqrt{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.7

Multiply

$4.4 \frac{v \sqrt{2}}{2}$ .

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Step 3.1.7.1

Combine

$4.4$ and

$\frac{v \sqrt{2}}{2}$ .

$\frac{4.4 (v \sqrt{2})}{2} \cdot \frac{2}{v \sqrt{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.7.2

Multiply

$\sqrt{2}$ by

$4.4$ .

$\frac{6.22253967 v}{2} \cdot \frac{2}{v \sqrt{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.8

Cancel the common factor of

$v$ .

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Step 3.1.8.1

Factor

$v$ out of

$6.22253967 v$ .

$\frac{v \cdot 6.22253967}{2} \cdot \frac{2}{v \sqrt{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.8.2

Factor

$v$ out of

$v \sqrt{2}$ .

$\frac{v \cdot 6.22253967}{2} \cdot \frac{2}{v (\sqrt{2})} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.8.3

Cancel the common factor.

$\frac{v \cdot 6.22253967}{2} \cdot \frac{2}{v \sqrt{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.8.4

Rewrite the expression.

$\frac{6.22253967}{2} \cdot \frac{2}{\sqrt{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.9

Cancel the common factor of

$2$ .

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Step 3.1.9.1

Cancel the common factor.

$\frac{6.22253967}{2} \cdot \frac{2}{\sqrt{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.9.2

Rewrite the expression.

$6.22253967 (\frac{1}{\sqrt{2}}) - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.10

Combine

$6.22253967$ and

$\frac{1}{\sqrt{2}}$ .

$\frac{6.22253967}{\sqrt{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.11

Multiply

$\frac{6.22253967}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{6.22253967}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.12

Combine and simplify the denominator.

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Step 3.1.12.1

Multiply

$\frac{6.22253967}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{6.22253967 \sqrt{2}}{\sqrt{2} \sqrt{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.12.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$\frac{6.22253967 \sqrt{2}}{\sqrt{2} \sqrt{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.12.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$\frac{6.22253967 \sqrt{2}}{\sqrt{2} \sqrt{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.12.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{6.22253967 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.12.5

Add

$1$ and

$1$ .

$\frac{6.22253967 \sqrt{2}}{{\sqrt{2}}^{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.12.6

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 3.1.12.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$\frac{6.22253967 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.12.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{6.22253967 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.12.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{6.22253967 \sqrt{2}}{2^{\frac{2}{2}}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.12.6.4

Cancel the common factor of

$2$ .

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Step 3.1.12.6.4.1

Cancel the common factor.

$\frac{6.22253967 \sqrt{2}}{2^{\frac{2}{2}}} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.12.6.4.2

Rewrite the expression.

$\frac{6.22253967 \sqrt{2}}{2} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.12.6.5

Evaluate the exponent.

$\frac{6.22253967 \sqrt{2}}{2} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.13

Multiply

$6.22253967$ by

$\sqrt{2}$ .

$\frac{8.8}{2} - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.14

Divide

$8.8$ by

$2$ .

$4.4 - 4.9 {(\frac{4.4}{v \cos (45)})}^{2} - 1.1 = 0$

Step 3.1.15

Separate fractions.

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{1}{\cos (45)})}^{2} - 1.1 = 0$

Step 3.1.16

Convert from

$\frac{1}{\cos (45)}$ to

$\sec (45)$ .

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \sec (45))}^{2} - 1.1 = 0$

Step 3.1.17

The exact value of

$\sec (45)$ is

$\frac{2}{\sqrt{2}}$ .

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2}{\sqrt{2}})}^{2} - 1.1 = 0$

Step 3.1.18

Multiply

$\frac{2}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$4.4 - 4.9 {(\frac{4.4}{v} \cdot (\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}))}^{2} - 1.1 = 0$

Step 3.1.19

Combine and simplify the denominator.

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Step 3.1.19.1

Multiply

$\frac{2}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2} - 1.1 = 0$

Step 3.1.19.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2} - 1.1 = 0$

Step 3.1.19.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2} - 1.1 = 0$

Step 3.1.19.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2} - 1.1 = 0$

Step 3.1.19.5

Add

$1$ and

$1$ .

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{2} - 1.1 = 0$

Step 3.1.19.6

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 3.1.19.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2} - 1.1 = 0$

Step 3.1.19.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2} - 1.1 = 0$

Step 3.1.19.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2} - 1.1 = 0$

Step 3.1.19.6.4

Cancel the common factor of

$2$ .

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Step 3.1.19.6.4.1

Cancel the common factor.

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2} - 1.1 = 0$

Step 3.1.19.6.4.2

Rewrite the expression.

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2 \sqrt{2}}{2})}^{2} - 1.1 = 0$

Step 3.1.19.6.5

Evaluate the exponent.

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2 \sqrt{2}}{2})}^{2} - 1.1 = 0$

Step 3.1.20

Cancel the common factor of

$2$ .

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Step 3.1.20.1

Cancel the common factor.

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \frac{2 \sqrt{2}}{2})}^{2} - 1.1 = 0$

Step 3.1.20.2

Divide

$\sqrt{2}$ by

$1$ .

$4.4 - 4.9 {(\frac{4.4}{v} \cdot \sqrt{2})}^{2} - 1.1 = 0$

Step 3.1.21

Multiply

$\frac{4.4}{v} \sqrt{2}$ .

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Step 3.1.21.1

Combine

$\frac{4.4}{v}$ and

$\sqrt{2}$ .

$4.4 - 4.9 {(\frac{4.4 \sqrt{2}}{v})}^{2} - 1.1 = 0$

Step 3.1.21.2

Multiply

$4.4$ by

$\sqrt{2}$ .

$4.4 - 4.9 {(\frac{6.22253967}{v})}^{2} - 1.1 = 0$

Step 3.1.22

Apply the product rule to

$\frac{6.22253967}{v}$ .

$4.4 - 4.9 \frac{{6.22253967}^{2}}{v^{2}} - 1.1 = 0$

Step 3.1.23

Raise

$6.22253967$ to the power of

$2$ .

$4.4 - 4.9 \frac{38.72}{v^{2}} - 1.1 = 0$

Step 3.1.24

Multiply

$- 4.9 \frac{38.72}{v^{2}}$ .

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Step 3.1.24.1

Combine

$- 4.9$ and

$\frac{38.72}{v^{2}}$ .

$4.4 + \frac{- 4.9 \cdot 38.72}{v^{2}} - 1.1 = 0$

Step 3.1.24.2

Multiply

$- 4.9$ by

$38.72$ .

$4.4 + \frac{- 189.728}{v^{2}} - 1.1 = 0$

Step 3.1.25

Move the negative in front of the fraction.

$4.4 - \frac{189.728}{v^{2}} - 1.1 = 0$

Step 3.2

Subtract

$1.1$ from

$4.4$ .

$- \frac{189.728}{v^{2}} + 3.3 = 0$

Step 4

Factor

$- \frac{189.728}{v^{2}} + 3.3$ .

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Step 4.1

Rewrite

$3.3$ as

${1.81659021}^{2}$ .

$- \frac{189.728}{v^{2}} + {1.81659021}^{2} = 0$

Step 4.2

Rewrite

$\frac{189.728}{v^{2}}$ as

${(\frac{13.77417874}{v})}^{2}$ .

$- {(\frac{13.77417874}{v})}^{2} + {1.81659021}^{2} = 0$

Step 4.3

Reorder

$- {(\frac{13.77417874}{v})}^{2}$ and

${1.81659021}^{2}$ .

${1.81659021}^{2} - {(\frac{13.77417874}{v})}^{2} = 0$

Step 4.4

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = 1.81659021$ and

$b = \frac{13.77417874}{v}$ .

$(1.81659021 + \frac{13.77417874}{v}) (1.81659021 - \frac{13.77417874}{v}) = 0$

Step 5

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$1.81659021 + \frac{13.77417874}{v} = 0$

$1.81659021 - \frac{13.77417874}{v} = 0$

Step 6

Set

$1.81659021 + \frac{13.77417874}{v}$ equal to

$0$ and solve for

$v$ .

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Step 6.1

Set

$1.81659021 + \frac{13.77417874}{v}$ equal to

$0$ .

$1.81659021 + \frac{13.77417874}{v} = 0$

Step 6.2

Solve

$1.81659021 + \frac{13.77417874}{v} = 0$ for

$v$ .

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Step 6.2.1

Subtract

$1.81659021$ from both sides of the equation.

$\frac{13.77417874}{v} = - 1.81659021$

Step 6.2.2

Find the LCD of the terms in the equation.

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Step 6.2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$v, 1$

Step 6.2.2.2

The LCM of one and any expression is the expression.

$v$

Step 6.2.3

Multiply each term in

$\frac{13.77417874}{v} = - 1.81659021$ by

$v$ to eliminate the fractions.

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Step 6.2.3.1

Multiply each term in

$\frac{13.77417874}{v} = - 1.81659021$ by

$v$ .

$\frac{13.77417874}{v} v = - 1.81659021 v$

Step 6.2.3.2

Simplify the left side.

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Step 6.2.3.2.1

Cancel the common factor of

$v$ .

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Step 6.2.3.2.1.1

Cancel the common factor.

$\frac{13.77417874}{v} v = - 1.81659021 v$

Step 6.2.3.2.1.2

Rewrite the expression.

$13.77417874 = - 1.81659021 v$

Step 6.2.4

Solve the equation.

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Step 6.2.4.1

Rewrite the equation as

$- 1.81659021 v = 13.77417874$ .

$- 1.81659021 v = 13.77417874$

Step 6.2.4.2

Divide each term in

$- 1.81659021 v = 13.77417874$ by

$- 1.81659021$ and simplify.

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Step 6.2.4.2.1

Divide each term in

$- 1.81659021 v = 13.77417874$ by

$- 1.81659021$ .

$\frac{- 1.81659021 v}{- 1.81659021} = \frac{13.77417874}{- 1.81659021}$

Step 6.2.4.2.2

Simplify the left side.

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Step 6.2.4.2.2.1

Cancel the common factor of

$- 1.81659021$ .

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Step 6.2.4.2.2.1.1

Cancel the common factor.

$\frac{- 1.81659021 v}{- 1.81659021} = \frac{13.77417874}{- 1.81659021}$

Step 6.2.4.2.2.1.2

Divide

$v$ by

$1$ .

$v = \frac{13.77417874}{- 1.81659021}$

Step 6.2.4.2.3

Simplify the right side.

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Step 6.2.4.2.3.1

Divide

$13.77417874$ by

$- 1.81659021$ .

$v = - 7.58243584$

Step 7

Set

$1.81659021 - \frac{13.77417874}{v}$ equal to

$0$ and solve for

$v$ .

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Step 7.1

Set

$1.81659021 - \frac{13.77417874}{v}$ equal to

$0$ .

$1.81659021 - \frac{13.77417874}{v} = 0$

Step 7.2

Solve

$1.81659021 - \frac{13.77417874}{v} = 0$ for

$v$ .

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Step 7.2.1

Subtract

$1.81659021$ from both sides of the equation.

$- \frac{13.77417874}{v} = - 1.81659021$

Step 7.2.2

Find the LCD of the terms in the equation.

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Step 7.2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$v, 1$

Step 7.2.2.2

The LCM of one and any expression is the expression.

$v$

Step 7.2.3

Multiply each term in

$- \frac{13.77417874}{v} = - 1.81659021$ by

$v$ to eliminate the fractions.

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Step 7.2.3.1

Multiply each term in

$- \frac{13.77417874}{v} = - 1.81659021$ by

$v$ .

$- \frac{13.77417874}{v} v = - 1.81659021 v$

Step 7.2.3.2

Simplify the left side.

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Step 7.2.3.2.1

Cancel the common factor of

$v$ .

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Step 7.2.3.2.1.1

Move the leading negative in

$- \frac{13.77417874}{v}$ into the numerator.

$\frac{- 13.77417874}{v} v = - 1.81659021 v$

Step 7.2.3.2.1.2

Cancel the common factor.

$\frac{- 13.77417874}{v} v = - 1.81659021 v$

Step 7.2.3.2.1.3

Rewrite the expression.

$- 13.77417874 = - 1.81659021 v$

Step 7.2.4

Solve the equation.

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Step 7.2.4.1

Rewrite the equation as

$- 1.81659021 v = - 13.77417874$ .

$- 1.81659021 v = - 13.77417874$

Step 7.2.4.2

Divide each term in

$- 1.81659021 v = - 13.77417874$ by

$- 1.81659021$ and simplify.

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Step 7.2.4.2.1

Divide each term in

$- 1.81659021 v = - 13.77417874$ by

$- 1.81659021$ .

$\frac{- 1.81659021 v}{- 1.81659021} = \frac{- 13.77417874}{- 1.81659021}$

Step 7.2.4.2.2

Simplify the left side.

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Step 7.2.4.2.2.1

Cancel the common factor of

$- 1.81659021$ .

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Step 7.2.4.2.2.1.1

Cancel the common factor.

$\frac{- 1.81659021 v}{- 1.81659021} = \frac{- 13.77417874}{- 1.81659021}$

Step 7.2.4.2.2.1.2

Divide

$v$ by

$1$ .

$v = \frac{- 13.77417874}{- 1.81659021}$

Step 7.2.4.2.3

Simplify the right side.

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Step 7.2.4.2.3.1

Divide

$- 13.77417874$ by

$- 1.81659021$ .

$v = 7.58243584$

Step 8

The final solution is all the values that make

$(1.81659021 + \frac{13.77417874}{v}) (1.81659021 - \frac{13.77417874}{v}) = 0$ true.

$v = - 7.58243584, 7.58243584$