Solve the equation 524 + 5x/113 + x = 4.66.

Step-by-step solution

1. 1

   Merge the two x terms over a common denominator. Writing $x$ as $\tfrac{113x}{113}$:

   $\frac{5x}{113} + x = \frac{5x + 113x}{113} = \frac{118}{113}\,x$

   so the equation becomes $524 + \tfrac{118}{113}x = 4.66$.

2. 2

   Move the constant across.

   $\frac{118}{113}x = 4.66 - 524 = -519.34$

   The right-hand side is large and negative because the constant $524$ dwarfs the target $4.66$ — that already tells us the root will be a few hundred below zero.

3. 3

   Multiply by the reciprocal.

   $x = -519.34 \times \frac{113}{118}$

4. 4

   Evaluate exactly, then round. Writing $519.34 = \tfrac{51934}{100} = \tfrac{25967}{50}$:

   $x = -\frac{25967 \times 113}{50 \times 118} = -\frac{2934271}{5900} \approx -497.334068$

   Doing the multiplication before the division keeps every digit; rounding $\tfrac{113}{118} \approx 0.9576$ first would land near $-497.37$, which is wrong in the second decimal.

5. 5

   Check by substitution. With $x = -497.334068$: $\tfrac{5x}{113} = -22.005932$, and $524 - 22.005932 - 497.334068 = 4.660000$. Using the wrongly rounded $-497.37$ instead gives $4.622$, off by nearly $0.04$.