Write a system of linear equations from the following situation and determine the solution. There are 24 animals in the field, some four-legged goats and some two-legged geese. There are a total number of 76 legs. How many of each type of animal are in the field?(1 point) Responses 10 geese and 14 goats 10 geese and 14 goats 12 geese and 12 goats 12 geese and 12 goats 14 geese and 10 goats 14 geese and 10 goats 11 geese and 13 goats

Answer 1

Let \( g \) represent the number of geese and \( t \) represent the number of goats. We can create a system of linear equations based on the information given:

1. The total number of animals (geese + goats) is 24: \[ g + t = 24 \]

2. The total number of legs (geese have 2 legs, and goats have 4 legs) is 76: \[ 2g + 4t = 76 \]

Now we can solve this system of equations.

Step 1: Solve the first equation for \( g \)

From the first equation: \[ g = 24 - t \]

Step 2: Substitute \( g \) in the second equation

Substituting \( g \) into the second equation: \[ 2(24 - t) + 4t = 76 \] This simplifies to: \[ 48 - 2t + 4t = 76 \] \[ 48 + 2t = 76 \] \[ 2t = 76 - 48 \] \[ 2t = 28 \] \[ t = 14 \]

Step 3: Find \( g \)

Now substituting \( t = 14 \) back into the first equation: \[ g + 14 = 24 \] \[ g = 24 - 14 \] \[ g = 10 \]

Conclusion

The number of geese is 10 and the number of goats is 14.

Thus, the solution is: 10 geese and 14 goats.