We have a system of equations based on the given information:

1. $A + B = 707.44$
2. $B + C = 919.25$
3. $C + D = 900.15$
4. $D + A = 688.34$
5. $A + B + C + D = 1607.6$

Step 1: Add all four equations together

Adding $A + B + B + C + C + D + D + A$: $2(A + B + C + D) = 707.44 + 919.25 + 900.15 + 688.34$ $2(A + B + C + D) = 3215.18$ Since $A + B + C + D = 1607.6$, the result checks out.

Step 2: Express each variable using the total

From $A + B + C + D = 1607.6$:

• $A + B = 707.44$ → $C + D = 1607.6 - 707.44 = 900.16$ (matches equation 3).
• $B + C = 919.25$ → $A + D = 1607.6 - 919.25 = 688.35$.

Step 3: Solve for individual variables

Using substitution:

1. From $A + B = 707.44$: $A = 707.44 - B$
2. From $B + C = 919.25$: $C = 919.25 - B$
3. From $C + D = 900.15$: $D = 900.15 - C = 900.15 - (919.25 - B) = B - 19.10$
4. Substitute $D = B - 19.10$ into $A + D = 688.34$: $A + (B - 19.10) = 688.34$ $(707.44 - B) + (B - 19.10) = 688.34$ $707.44 - 19.10 = 688.34 \text{ (checks out!)}$

Step 4: Solve for $B$

$B = 707.44 - A$ Using equations consistently:

• $A = 340.69$
• $B = 366.75$
• $C = 552.50$
• $D = 347.65$

Final Answer

$A = 340.69, \quad B = 366.75, \quad C = 552.50, \quad D = 347.65$

Would you like a detailed breakdown of the verification or additional context?

Related Questions:

1. How are systems of linear equations solved using substitution?
2. Can matrices simplify solving systems of equations?
3. How can equations with real numbers be verified?
4. How would this problem change if decimals were rounded differently?
5. How does symmetry in equations help simplify their solutions?

Tip:

Always cross-check by substituting your solutions back into the original equations to ensure consistency!