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Solution - Linear equations with one unknown

u = 1

u=1

See steps

Step-by-step explanation

1. Simplify the expression

$\frac{3}{5} \cdot (\frac{1}{2} \cdot u + 6 - 5 u) = \frac{9}{10} u$

Group like terms:

$\frac{3}{5} \cdot ((\frac{1}{2} \cdot u - 5 u) + 6) = \frac{9}{10} u$

Group the coefficients:

$\frac{3}{5} \cdot ((\frac{1}{2} - 5) u + 6) = \frac{9}{10} u$

Convert the integer into a fraction:

$\frac{3}{5} \cdot ((\frac{1}{2} + \frac{- 10}{2}) u + 6) = \frac{9}{10} u$

Combine the fractions:

$\frac{3}{5} \cdot (\frac{(1 - 10)}{2} \cdot u + 6) = \frac{9}{10} u$

Combine the numerators:

$\frac{3}{5} \cdot (\frac{- 9}{2} \cdot u + 6) = \frac{9}{10} u$

Multiply the fraction(s):

$\frac{(3 \cdot (\frac{- 9}{2} \cdot u + 6))}{5} = \frac{9}{10} u$

Expand the parentheses:

$\frac{(3 \cdot \frac{- 9}{2} \cdot u + 3 \cdot 6)}{5} = \frac{9}{10} u$

Multiply the coefficients:

$\frac{(\frac{(3 \cdot - 9)}{2} \cdot u + 3 \cdot 6)}{5} = \frac{9}{10} u$

Simplify the arithmetic:

$\frac{(\frac{(3 \cdot - 9)}{2} \cdot u + 18)}{5} = \frac{9}{10} u$

$\frac{(\frac{- 27}{2} \cdot u + 18)}{5} = \frac{9}{10} u$

Break up the fraction:

$\frac{\frac{- 27}{2} u}{5} + \frac{18}{5} = \frac{9}{10} u$

Swap sides:

$\frac{9}{10} u = \frac{\frac{- 27}{2} u}{5} + \frac{18}{5}$

2. Isolate the u

$\frac{9}{10} \cdot u = \frac{\frac{- 27}{2} u}{5} + \frac{18}{5}$

Multiply both sides by inverse fraction $\frac{10}{9}$ :

$(\frac{9}{10} u) \cdot \frac{10}{9} = (\frac{\frac{- 27}{2} u}{5} + \frac{18}{5}) \cdot \frac{10}{9}$

Group like terms:

$(\frac{9}{10} \cdot \frac{10}{9}) u = (\frac{\frac{- 27}{2} u}{5} + \frac{18}{5}) \cdot \frac{10}{9}$

Multiply the coefficients:

$\frac{(9 \cdot 10)}{(10 \cdot 9)} \cdot u = (\frac{\frac{- 27}{2} u}{5} + \frac{18}{5}) \cdot \frac{10}{9}$

Simplify the fraction:

$u = (\frac{\frac{- 27}{2} u}{5} + \frac{18}{5}) \cdot \frac{10}{9}$

Multiply the fraction(s):

$u = \frac{((\frac{\frac{- 27}{2} u}{5} + \frac{18}{5}) \cdot 10)}{9}$

Expand the parentheses:

$u = \frac{(\frac{(\frac{- 27}{2} u \cdot 10)}{5} + \frac{(18 \cdot 10)}{5})}{9}$

Group like terms:

$u = \frac{(\frac{((\frac{- 27}{2} \cdot 10) u)}{5} + \frac{(18 \cdot 10)}{5})}{9}$

Multiply the coefficients:

$u = \frac{(\frac{(\frac{(- 27 \cdot 10)}{2} u)}{5} + \frac{(18 \cdot 10)}{5})}{9}$

Simplify the arithmetic:

$u = \frac{(\frac{(\frac{(- 27 \cdot 10)}{2} u)}{5} + \frac{180}{5})}{9}$

Find the greatest common factor of the numerator and denominator:

$u = \frac{(\frac{((- 27 \cdot 5) u)}{5} + \frac{(36 \cdot 5)}{(1 \cdot 5)})}{9}$

Factor out and cancel the greatest common factor:

$u = \frac{(\frac{((- 27 \cdot 5) u)}{5} + 36)}{9}$

Simplify the arithmetic:

$u = \frac{(\frac{- 135 u}{5} + 36)}{9}$

Simplify the fraction:

$u = \frac{(- 27 u + 36)}{9}$

Break up the fraction:

$u = \frac{- 27 u}{9} + \frac{36}{9}$

Simplify the fraction:

$u = - 3 u + \frac{36}{9}$

Find the greatest common factor of the numerator and denominator:

$u = - 3 u + \frac{(4 \cdot 9)}{(1 \cdot 9)}$

Factor out and cancel the greatest common factor:

$u = - 3 u + 4$

3. Group all u terms on the left side of the equation

$u = - 3 u + 4$

Add $3 u$ to both sides:

$u + 3 u = (- 3 u + 4) + 3 u$

Simplify the arithmetic:

$4 u = (- 3 u + 4) + 3 u$

Group like terms:

$4 u = (- 3 u + 3 u) + 4$

Simplify the arithmetic:

$4 u = 4$

4. Isolate the u

$4 u = 4$

Divide both sides by 4:

$\frac{(4 u)}{4} = \frac{4}{4}$

Simplify the fraction:

$u = \frac{4}{4}$

Simplify the fraction:

$u = 1$

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Why learn this

Linear equations cannot tell you the future, but they can give you a good idea of what to expect so you can plan ahead. How long will it take you to fill your swimming pool? How much money will you earn during summer break? What are the quantities you need for your favorite recipe to make enough for all your friends?

Linear equations explain some of the relationships between what we know and what we want to know and can help us solve a wide range of problems we might encounter in our everyday lives.

Terms and topics

Linear equations with one unknown

Solution - Linear equations with one unknown

Step-by-step explanation

1. Simplify the expression

2. Isolate the u

3. Group all u terms on the left side of the equation

4. Isolate the u

Why learn this

Terms and topics

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