Solve for x: 122,400,000 + 0.72x + 12x + 1,020,000 + 400,000 + 0.13 × (122,400,000 − 1200x) = 1200x.

Step-by-step solution

1. 1

   Combine the plain constants and the two simple $x$-terms.

   $0.72x+12x=12.72x,\qquad 122{,}400{,}000+1{,}020{,}000+400{,}000=123{,}820{,}000,$

   so the equation becomes

   $123{,}820{,}000+12.72x+0.13\bigl(122{,}400{,}000-1200x\bigr)=1200x.$

2. 2

   Distribute the 13%.

   $0.13(122{,}400{,}000)=15{,}912{,}000,\qquad 0.13(1200x)=156x,$

   so that bracket contributes $15{,}912{,}000-156x$. The $-156x$ is large enough to reverse the sign of the whole $x$-coefficient on the left, which is the feature that makes this equation interesting rather than routine.

3. 3

   Collect the left-hand side.

   $123{,}820{,}000+15{,}912{,}000=139{,}732{,}000,\qquad 12.72x-156x=-143.28x,$

   giving

   $139{,}732{,}000-143.28x=1200x.$

4. 4

   Bring the $x$-terms together. Because the left coefficient is negative, adding $143.28x$ to both sides raises the right-hand coefficient rather than lowering it:

   $139{,}732{,}000=1200x+143.28x=1343.28x.$

5. 5

   Divide and give the exact value.

   $x=\frac{139{,}732{,}000}{1343.28}=\frac{9{,}050{,}000}{87}=104{,}022.9885057\ldots$

   Substituting back with exact fractions makes both sides equal $124{,}827{,}586.20689656$. A slip in the final division that gives $104{,}025.97$ is off by about $3$ units — worth catching, since $1343.28\times104{,}025.97=139{,}736{,}000$, not $139{,}732{,}000$.