Solve the system D + W = 25000 and 0.96D + 0.015W = 12500 for D and W.

Step-by-step solution

1. 1

   Simplify the right-hand equation and read the situation. Since $25000\times 0.5=12500$, the system is

   $D+W=25000,\qquad 0.96D+0.015W=12500$

   This is a mixture: two streams with concentrations $96\%$ and $1.5\%$ combine to $25000$ units at $50\%$. Because $50\%$ lies between the two rates, a sensible positive solution should exist.

2. 2

   Solve the simple equation for one variable. The first equation has coefficients of 1, so isolating is free:

   $W=25000-D$

   Substitution beats elimination here precisely because one equation is already this clean.

3. 3

   Substitute into the second equation.

   $0.96D+0.015(25000-D)=12500$

   $0.96D+375-0.015D=12500$

   Here $0.015\times 25000=375$ — the contribution the weak stream would make on its own if it filled the whole tank.

4. 4

   Combine and solve for $D$. The $D$ coefficient is the difference of the two rates:

   $(0.96-0.015)D=0.945D,\qquad 0.945D+375=12500$

   $0.945D=12125\ \Longrightarrow\ D=\frac{12125}{0.945}=12830.6878\ldots\approx 12830.69$

5. 5

   Back-substitute for $W$.

   $W=25000-12830.69=12169.31$

   Both values are positive and each is less than the total, as a physical mixture requires.

6. 6

   Check both original equations. Sum: $12830.69+12169.31=25000$ ✓. Weighted sum: $0.96\times 12830.69+0.015\times 12169.31=12317.46+182.54=12500.00$ ✓. Since the two checks are independent, passing both confirms the pair.