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Find out how many chickens a farmer has when they also keep some cows. There are a total of 115 animals and 304 legs on the farm.
A Chicken Farmer also has some cows for a total of 115 animals, and the animals have 304 legs in all. How many Chickens does the Farmer have?
The Farmer has 78 Chickens.
Let’s denote the number of chickens as C and the number of cows as K.
- The total number of animals is 115: C + K = 115
- The total number of legs is 304. Since each chicken has 2 legs and each cow has 4 legs: 2C + 4K = 304
We can solve these two equations simultaneously to find the values of C and K.
First, let’s express one variable in terms of the other in the first equation: K = 115 – C
Now, substitute this expression for K into the second equation: 2C + 4(115 – C) = 304
2C + 460 – 4C = 304
-2C + 460 = 304
-2C = 304 – 460
-2C = -156
Divide both sides by -2: C = -156 / -2
C = 78
So, the farmer has 78 chickens and 37 cows.
System of Linear Equations in Algebra
Systems of linear equations are a fundamental concept in algebra, and they have many applications in various fields. Here’s a breakdown:
What are they?
- A system of linear equations is a collection of two or more equations involving the same variables.
- Each equation is linear, meaning it has the form ax + by + cz = d, where a, b, and c are constants, and x, y, and z are the variables.
- The goal is to find the values of the variables that make all the equations true simultaneously.
Key terms:
- Solution: A set of values for the variables that satisfies all the equations in the system.
- Solution set: The collection of all possible solutions to the system.
- Consistent system: A system that has at least one solution.
- Inconsistent system: A system that has no solution.
Examples:
- A system with two variables and two equations:
- A system with three variables and three equations:
- w + 2x – y = 4
- 2w + 3x + y = 8
- w – x + 2y = 3
Solving systems:
There are various methods for solving systems of linear equations, depending on the size and complexity of the system. Some common methods include:
- Elimination: Eliminating one variable from one or more equations by adding or subtracting them strategically.
- Substitution: Solving one equation for one variable and substituting that expression into another equation.
- Gaussian elimination: Converting the system to an augmented matrix and performing row operations to reduce it to a triangular form, from which the solutions can be easily obtained.
- Matrix methods: Using matrix operations to solve the system in a more systematic way.
Applications:
Systems of linear equations are used in various fields, including:
- Physics: Modeling physical systems like forces, motion, and heat flow.
- Chemistry: Balancing chemical reactions and analyzing chemical mixtures.
- Economics: Modeling economic systems like supply and demand.
- Computer science: Solving optimization problems and designing algorithms.
- Engineering: Designing structures, analyzing circuits, and controlling systems.
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