Farm Animals Word Problem: Ducks and Cows

This problem involves finding the number of ducks given the total number of heads and legs for ducks and cows on a farm. We can solve this using a system of linear equations.

Setting up the Equations

Let $d$ be the number of ducks and $c$ be the number of cows.

• Each animal has one head. The total number of heads is 7.
  Equation 1 (Heads): $ d + c = 7 $
• Ducks have 2 legs and cows have 4 legs. The total number of legs is 18.
  Equation 2 (Legs): $ 2d + 4c = 18 $

Solving the System

We can use substitution to find the value of $d$.

1. Solve Equation 1 for $c$:
   $ c = 7 - d $
2. Substitute this expression for $c$ into Equation 2:
   $ 2d + 4(7 - d) = 18 $
3. Simplify the equation:
   $ 2d + 28 - 4d = 18 $
   $ -2d + 28 = 18 $
4. Isolate the term with $d$:
   $ -2d = 18 - 28 $
   $ -2d = -10 $
5. Solve for $d$:
   $ d = \frac{-10}{-2} $
   $ d = 5 $

Conclusion

The calculation shows there are 5 ducks.

Verification: If $d = 5$, then $c = 7 - 5 = 2$. Total legs = $(5 \times 2) + (2 \times 4) = 10 + 8 = 18$. This matches the given information.