Solve for x: [178.29 + (1024.32 - x)] / (1279.43 + x) = 16/9.

Step-by-step solution

1. 1

   Tidy the numerator before doing anything structural. The two constants combine:

   $178.29+(1024.32-x)=1202.61-x$

   so the equation is $\dfrac{1202.61-x}{1279.43+x}=\dfrac{16}{9}$. Note that $x$ appears in both numerator and denominator with opposite signs, which is what makes the final coefficient large.

2. 2

   Cross-multiply. For $\dfrac{A}{B}=\dfrac{C}{D}$ with $B,D\neq 0$, the equation $AD=BC$ is equivalent:

   $9(1202.61-x)=16(1279.43+x)$

   Keep in mind the hidden restriction $1279.43+x\neq 0$, i.e. $x\neq -1279.43$; we will check the answer against it at the end.

3. 3

   Expand both sides.

   $9\times 1202.61=10823.49,\qquad 16\times 1279.43=20470.88$

   $10823.49-9x=20470.88+16x$

4. 4

   Collect the $x$ terms and the constants. Subtracting $10823.49$ and $16x$ from both sides:

   $-25x=20470.88-10823.49=9647.39$

   The coefficient $-25$ comes from $-9-16$: because $x$ sat on opposite sides of the fraction bar, the two contributions add in magnitude instead of partially cancelling.

5. 5

   Divide and interpret the sign.

   $x=\frac{9647.39}{-25}=-385.8956$

   It is exact, not rounded, since $9647.39/25$ terminates. The value differs from $-1279.43$, so the denominator is not zero and the solution is admissible.

6. 6

   Check by substituting back. Numerator: $1202.61-(-385.8956)=1588.5056$. Denominator: $1279.43+(-385.8956)=893.5344$. Their ratio is $1588.5056/893.5344=1.7\overline{7}=16/9$ ✓.