Solve the system: X * 5.15 + Y * 5.70 = 8615.35 and X + Y = 1555.

Step-by-step solution

1. 1

   Choose elimination and decide which variable to remove. The second equation has coefficient $1$ on both unknowns, so scaling it is cheap. Multiplying it by $5.15$ makes the $X$ coefficients match:

   $5.15X+5.15Y=5.15\cdot 1555=8008.25$

2. 2

   Subtract to eliminate $X$.

   $\left(5.15X+5.70Y\right)-\left(5.15X+5.15Y\right)=8615.35-8008.25$

   $0.55Y=607.10$

   The coefficient $0.55=5.70-5.15$ is the price difference, which is why this method is so natural for two-price problems.

3. 3

   Solve for $Y$ and keep it exact.

   $Y=\frac{607.10}{0.55}=\frac{60710}{55}=\frac{12142}{11}=1103.\overline{81}$

   Converting to a fraction avoids rounding drift; the decimal $1103.82$ is only an approximation.

4. 4

   Back-substitute for $X$.

   $X=1555-\frac{12142}{11}=\frac{17105-12142}{11}=\frac{4963}{11}=451.\overline{18}$

5. 5

   Check both equations exactly. The sum: $\tfrac{4963}{11}+\tfrac{12142}{11}=\tfrac{17105}{11}=1555\;\checkmark$. The value equation, cleared of decimals by multiplying through by $100$:

   $\frac{515\cdot 4963+570\cdot 12142}{11}=\frac{2555945+6920940}{11}=\frac{9476885}{11}=861535$

   which is $8615.35\times 100\;\checkmark$. Note that a rounded pair such as $X\approx 445.7$, $Y\approx 1109.3$ fails this check — it gives $8618.35$, three units too high.