Solve for x: 10,225 + 36,954.76x + 277.94x + 227.39x = 883,466 + 7,067.9x + 292,043 + 0.30 × (10,225 + 36,954.76x + 277.94x + 227.39x) + 333,025.

Step-by-step solution

1. 1

   Combine the three $x$-rates that appear on both sides.

   $36{,}954.76+277.94+227.39=37{,}460.09,$

   so the repeated block is

   $L=10{,}225+37{,}460.09x,$

   and the equation reads $L=1{,}508{,}534+7{,}067.9x+0.30L$, where $883{,}466+292{,}043+333{,}025=1{,}508{,}534$.

2. 2

   Move the $0.30L$ term to the left.

   $0.70L=1{,}508{,}534+7{,}067.9x.$

   Equivalently, expand it in place: $0.30(10{,}225+37{,}460.09x)=3{,}067.5+11{,}238.027x$, which is the form most textbooks use.

3. 3

   Write both sides out and collect. With the expansion,

   $10{,}225+37{,}460.09x=1{,}511{,}601.5+18{,}305.927x,$

   since $1{,}508{,}534+3{,}067.5=1{,}511{,}601.5$ and $7{,}067.9+11{,}238.027=18{,}305.927$.

4. 4

   Separate the unknown from the constants.

   $37{,}460.09x-18{,}305.927x=1{,}511{,}601.5-10{,}225\;\Longrightarrow\;19{,}154.163x=1{,}501{,}376.5.$

   The net rate $19{,}154.163=0.70(37{,}460.09)-7{,}067.9$ is the margin per unit after the $30\%$ share.

5. 5

   Divide, and carry enough digits.

   $x=\frac{1{,}501{,}376.5}{19{,}154.163}=\frac{1{,}501{,}376{,}500}{19{,}154{,}163}=78.3838218355\ldots$

   Substituting the exact fraction back makes both sides equal $2{,}946{,}490.0205015484$. Note the answer is $78.38382$, not $78.38330$ — truncating the division too early costs the fourth decimal place.