Solve T=(1/2+1/3+1/4)(1/2+frac{1}{3}+frac{1}{4}-frac{1}{5}-frac{1}{6}+1)+(1-frac{1}{2}-frac{1}{3}-frac{1}{4})(frac{1}{2}+frac{1}{3}+frac{1}{4}-frac{1}{5}-frac{1}{6})

$$T = ( \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } ) ( \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } - \frac { 1 } { 5 } - \frac { 1 } { 6 } + 1 ) + ( 1 - \frac { 1 } { 2 } - \frac { 1 } { 3 } - \frac { 1 } { 4 } ) ( \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } - \frac { 1 } { 5 } - \frac { 1 } { 6 } )$$

Solve for T

$$T = \frac{9}{5} = 1\frac{4}{5} = 1.8$$

Steps for Solving Linear Equation

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{1}{3} to fractions with denominator 6.

$$T=\left(\frac{3}{6}+\frac{2}{6}+\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Since \frac{3}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$T=\left(\frac{3+2}{6}+\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Add 3 and 2 to get 5.

$$T=\left(\frac{5}{6}+\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Least common multiple of 6 and 4 is 12. Convert \frac{5}{6} and \frac{1}{4} to fractions with denominator 12.

$$T=\left(\frac{10}{12}+\frac{3}{12}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Since \frac{10}{12} and \frac{3}{12} have the same denominator, add them by adding their numerators.

$$T=\frac{10+3}{12}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Add 10 and 3 to get 13.

$$T=\frac{13}{12}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{1}{3} to fractions with denominator 6.

$$T=\frac{13}{12}\left(\frac{3}{6}+\frac{2}{6}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Since \frac{3}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$T=\frac{13}{12}\left(\frac{3+2}{6}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Add 3 and 2 to get 5.

$$T=\frac{13}{12}\left(\frac{5}{6}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Least common multiple of 6 and 4 is 12. Convert \frac{5}{6} and \frac{1}{4} to fractions with denominator 12.

$$T=\frac{13}{12}\left(\frac{10}{12}+\frac{3}{12}-\frac{1}{5}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Since \frac{10}{12} and \frac{3}{12} have the same denominator, add them by adding their numerators.

$$T=\frac{13}{12}\left(\frac{10+3}{12}-\frac{1}{5}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Add 10 and 3 to get 13.

$$T=\frac{13}{12}\left(\frac{13}{12}-\frac{1}{5}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Least common multiple of 12 and 5 is 60. Convert \frac{13}{12} and \frac{1}{5} to fractions with denominator 60.

$$T=\frac{13}{12}\left(\frac{65}{60}-\frac{12}{60}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Since \frac{65}{60} and \frac{12}{60} have the same denominator, subtract them by subtracting their numerators.

$$T=\frac{13}{12}\left(\frac{65-12}{60}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Subtract 12 from 65 to get 53.

$$T=\frac{13}{12}\left(\frac{53}{60}-\frac{1}{6}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Least common multiple of 60 and 6 is 60. Convert \frac{53}{60} and \frac{1}{6} to fractions with denominator 60.

$$T=\frac{13}{12}\left(\frac{53}{60}-\frac{10}{60}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Since \frac{53}{60} and \frac{10}{60} have the same denominator, subtract them by subtracting their numerators.

$$T=\frac{13}{12}\left(\frac{53-10}{60}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Subtract 10 from 53 to get 43.

$$T=\frac{13}{12}\left(\frac{43}{60}+1\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Convert 1 to fraction \frac{60}{60}.

$$T=\frac{13}{12}\left(\frac{43}{60}+\frac{60}{60}\right)+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Since \frac{43}{60} and \frac{60}{60} have the same denominator, add them by adding their numerators.

$$T=\frac{13}{12}\times \frac{43+60}{60}+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Add 43 and 60 to get 103.

$$T=\frac{13}{12}\times \frac{103}{60}+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Multiply \frac{13}{12} times \frac{103}{60} by multiplying numerator times numerator and denominator times denominator.

$$T=\frac{13\times 103}{12\times 60}+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Do the multiplications in the fraction \frac{13\times 103}{12\times 60}.

$$T=\frac{1339}{720}+\left(1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Convert 1 to fraction \frac{2}{2}.

$$T=\frac{1339}{720}+\left(\frac{2}{2}-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, subtract them by subtracting their numerators.

$$T=\frac{1339}{720}+\left(\frac{2-1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Subtract 1 from 2 to get 1.

$$T=\frac{1339}{720}+\left(\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{1}{3} to fractions with denominator 6.

$$T=\frac{1339}{720}+\left(\frac{3}{6}-\frac{2}{6}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Since \frac{3}{6} and \frac{2}{6} have the same denominator, subtract them by subtracting their numerators.

$$T=\frac{1339}{720}+\left(\frac{3-2}{6}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Subtract 2 from 3 to get 1.

$$T=\frac{1339}{720}+\left(\frac{1}{6}-\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Least common multiple of 6 and 4 is 12. Convert \frac{1}{6} and \frac{1}{4} to fractions with denominator 12.

$$T=\frac{1339}{720}+\left(\frac{2}{12}-\frac{3}{12}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Since \frac{2}{12} and \frac{3}{12} have the same denominator, subtract them by subtracting their numerators.

$$T=\frac{1339}{720}+\frac{2-3}{12}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Subtract 3 from 2 to get -1.

$$T=\frac{1339}{720}-\frac{1}{12}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{1}{3} to fractions with denominator 6.

$$T=\frac{1339}{720}-\frac{1}{12}\left(\frac{3}{6}+\frac{2}{6}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Since \frac{3}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$T=\frac{1339}{720}-\frac{1}{12}\left(\frac{3+2}{6}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Add 3 and 2 to get 5.

$$T=\frac{1339}{720}-\frac{1}{12}\left(\frac{5}{6}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)$$

Least common multiple of 6 and 4 is 12. Convert \frac{5}{6} and \frac{1}{4} to fractions with denominator 12.

$$T=\frac{1339}{720}-\frac{1}{12}\left(\frac{10}{12}+\frac{3}{12}-\frac{1}{5}-\frac{1}{6}\right)$$

Since \frac{10}{12} and \frac{3}{12} have the same denominator, add them by adding their numerators.

$$T=\frac{1339}{720}-\frac{1}{12}\left(\frac{10+3}{12}-\frac{1}{5}-\frac{1}{6}\right)$$

Add 10 and 3 to get 13.

$$T=\frac{1339}{720}-\frac{1}{12}\left(\frac{13}{12}-\frac{1}{5}-\frac{1}{6}\right)$$

Least common multiple of 12 and 5 is 60. Convert \frac{13}{12} and \frac{1}{5} to fractions with denominator 60.

$$T=\frac{1339}{720}-\frac{1}{12}\left(\frac{65}{60}-\frac{12}{60}-\frac{1}{6}\right)$$

Since \frac{65}{60} and \frac{12}{60} have the same denominator, subtract them by subtracting their numerators.

$$T=\frac{1339}{720}-\frac{1}{12}\left(\frac{65-12}{60}-\frac{1}{6}\right)$$

Subtract 12 from 65 to get 53.

$$T=\frac{1339}{720}-\frac{1}{12}\left(\frac{53}{60}-\frac{1}{6}\right)$$

Least common multiple of 60 and 6 is 60. Convert \frac{53}{60} and \frac{1}{6} to fractions with denominator 60.

$$T=\frac{1339}{720}-\frac{1}{12}\left(\frac{53}{60}-\frac{10}{60}\right)$$

Since \frac{53}{60} and \frac{10}{60} have the same denominator, subtract them by subtracting their numerators.

$$T=\frac{1339}{720}-\frac{1}{12}\times \frac{53-10}{60}$$

Subtract 10 from 53 to get 43.

$$T=\frac{1339}{720}-\frac{1}{12}\times \frac{43}{60}$$

Multiply -\frac{1}{12} times \frac{43}{60} by multiplying numerator times numerator and denominator times denominator.

$$T=\frac{1339}{720}+\frac{-43}{12\times 60}$$

Do the multiplications in the fraction \frac{-43}{12\times 60}.

$$T=\frac{1339}{720}+\frac{-43}{720}$$

Fraction \frac{-43}{720} can be rewritten as -\frac{43}{720} by extracting the negative sign.

$$T=\frac{1339}{720}-\frac{43}{720}$$

Since \frac{1339}{720} and \frac{43}{720} have the same denominator, subtract them by subtracting their numerators.

$$T=\frac{1339-43}{720}$$

Subtract 43 from 1339 to get 1296.

$$T=\frac{1296}{720}$$

Reduce the fraction \frac{1296}{720} to lowest terms by extracting and canceling out 144.

$$T=\frac{9}{5}$$

Assign T

$$T≔\frac{9}{5}$$