7 heads and 24 legs - how many geese and cats are there? - briefly
This problem is a classic example of a system of linear equations, where one must determine the number of two types of animals—geese and cats—based on the total number of heads and legs. Geese have two legs each, while cats have four. To solve this, let's denote the number of geese as G and the number of cats as C.
Given the total number of heads is 7, we can write the first equation as: G + C = 7
Since geese have 2 legs and cats have 4, the total number of legs is 24. This gives us the second equation: 2G + 4C = 24
To solve these equations, first simplify the second equation by dividing everything by 2: G + 2C = 12
Now, subtract the first equation from this simplified equation: (G + 2C) - (G + C) = 12 - 7 C = 5
Substitute C back into the first equation to find G: G + 5 = 7 G = 2
There are 2 geese and 5 cats.
7 heads and 24 legs - how many geese and cats are there? - in detail
The problem of determining the number of geese and cats given 7 heads and 24 legs is a classic example of a system of linear equations. This type of problem is often used to illustrate the principles of algebra and to demonstrate how mathematical reasoning can be applied to solve real-world puzzles. Let's delve into the details of how to approach and solve this problem.
First, we need to establish the basic facts about geese and cats. Geese have 1 head and 2 legs each, while cats have 1 head and 4 legs each. This information is crucial for setting up the equations that will help us find the solution.
Let's denote the number of geese as G and the number of cats as C. We can then set up the following system of equations based on the given information:
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The total number of heads is 7. Since each goose and each cat has 1 head, we can write the equation as: G + C = 7
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The total number of legs is 24. Since geese have 2 legs and cats have 4 legs, we can write the equation as: 2G + 4C = 24
Now, we have a system of two equations with two variables. To solve for G and C, we can use the substitution or elimination method. Let's use the substitution method for clarity.
From the first equation, we can express C in terms of G: C = 7 - G
Next, we substitute this expression for C into the second equation: 2G + 4(7 - G) = 24
Simplify and solve for G: 2G + 28 - 4G = 24 -2G = -4 G = 2
Now that we have the value of G, we can find C by substituting G back into the equation C = 7 - G: C = 7 - 2 C = 5
Therefore, the solution to the problem is that there are 2 geese and 5 cats. This result is derived through a logical and systematic approach to solving a system of linear equations, highlighting the importance of mathematical reasoning in problem-solving.