Solve for x: 883,466 + 687,904 + 25.92x = 41,918 + 378,347 + 392.4x + 223,021 + 0.05 × (883,466 + 687,904 + 25.92x).

Step-by-step solution

1. 1

   Name the repeated block. The expression $883{,}466+687{,}904+25.92x$ appears twice, once on each side. Write

   $L=1{,}571{,}370+25.92x,$

   so the equation is $L=643{,}286+392.4x+0.05L$, where $41{,}918+378{,}347+223{,}021=643{,}286$. Spotting the repetition turns a sprawling equation into a short one.

2. 2

   Move the $0.05L$ to the left.

   $L-0.05L=0.95L=643{,}286+392.4x.$

   Equivalently, only $95\%$ of the left-hand block is available after the $5\%$ deduction.

3. 3

   Expand $0.95L$.

   $0.95\bigl(1{,}571{,}370+25.92x\bigr)=1{,}492{,}801.5+24.624x,$

   so

   $1{,}492{,}801.5+24.624x=643{,}286+392.4x.$

4. 4

   Separate constants and $x$-terms.

   $1{,}492{,}801.5-643{,}286=392.4x-24.624x\;\Longrightarrow\;849{,}515.5=367.776x.$

   The coefficient $367.776=392.4-0.95(25.92)$ is the net cost per unit once the commission is accounted for.

5. 5

   Divide and verify.

   $x=\frac{849{,}515.5}{367.776}=\frac{212{,}378{,}875}{91{,}944}=2309.8720417\ldots$

   Substituting back with exact fractions, both sides equal $1{,}631{,}241.883320282$, so the solution is exact rather than merely close.