Solve for X: (217*22 + (293 - 32X)*10) / (3.6*7*60) = (0.857X - 217/25.2)*28/60 + (0.857X - 103/25.2)*35/60 + (0.857X - 32X/25.2)*10/60.

Step-by-step solution

1. 1

   Simplify the left-hand numerator. $217\cdot 22=4774$ and $(293-32X)\cdot 10=2930-320X$, so the numerator is $7704-320X$. Everything to the left of the equals sign is linear in $X$.

2. 2

   Evaluate the left-hand denominator carefully. $3.6\cdot 7=25.2$ and $25.2\cdot 60=1512$. This is the one place where an arithmetic slip changes the whole answer, so it is worth doing in two explicit multiplications. The left side is therefore $\frac{7704-320X}{1512}=\frac{107}{21}-\frac{40}{189}X\approx 5.095238-0.211640X$.

3. 3

   Expand the three right-hand terms. Using $25.2\cdot 60=1512$: the constant parts are $-\frac{217}{25.2}\cdot\frac{28}{60}=-\frac{6076}{1512}=-4.018519$ and $-\frac{103}{25.2}\cdot\frac{35}{60}=-\frac{3605}{1512}=-2.384259$, summing to $-\frac{9681}{1512}=-\frac{461}{72}\approx -6.402778$. The third term has no constant part.

4. 4

   Collect the coefficient of X on the right. The first two terms contribute $0.857\left(\frac{28}{60}+\frac{35}{60}\right)=0.857\cdot 1.05=0.89985$, and the third contributes $\left(0.857-\frac{32}{25.2}\right)\frac{10}{60}=(0.857-1.269841)\cdot\frac16=-0.068807$. Together the coefficient is $\frac{3141343}{3780000}\approx 0.831043$.

5. 5

   Move all X terms to one side and constants to the other. From $5.095238-0.211640X=0.831043X-6.402778$ we get $5.095238+6.402778=(0.831043+0.211640)X$, that is $11.498016=1.042683X$.

6. 6

   Divide to finish. $X=\frac{11.498016}{1.042683}=\frac{14487500}{1313781}\approx 11.0273$.

7. 7

   Numerical check at X = 11.0273326. The left side is $\frac{7704-320(11.0273326)}{1512}=\frac{4175.253}{1512}=2.761411$, and the right side is $0.831043(11.0273326)-6.402778=2.761411$. Both sides agree to six decimal places.