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In a zoo, There are rabbits and pigeons. If heads are counted, there are 200 and if legs are counted, there are 580. How many pigeons are there?
There are 110 pigeons in the zoo.
Let’s denote the number of rabbits as R and the number of pigeons as P.
Each rabbit has 1 head and 4 legs, and each pigeon has 1 head and 2 legs.
From the information given, we can create two equations based on the number of heads and legs:
- The total number of heads: R + P = 200 (equation 1)
- The total number of legs: 4R + 2P = 580 (equation 2)
Now, we can solve this system of equations to find the values of R and P.
From equation 1, we can express R in terms of P: R = 200 – P.
Substituting this expression for R into equation 2, we get:
4(200 – P) + 2P = 580
800 – 4P + 2P = 580
800 – 2P = 580
2P = 800 – 580
2P = 220
P = 110
So, there are 110 pigeons in the zoo.
System of Linear Equations in Algebra
Systems of linear equations are a fundamental concept in algebra, and they have numerous applications in various fields like physics, economics, engineering, and computer science. Here’s an overview:
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What is a system of linear equations?
It’s a collection of two or more linear equations involving the same variables. Each equation represents a line in a coordinate system (for two variables) or a plane (for three variables). The solution to the system is the set of values for the variables that make all the equations true simultaneously.
Examples:
Solving systems of linear equations:
There are various methods for solving systems, depending on the size and complexity. Here are some common ones:
- Substitution: Solve one equation for a variable and substitute that expression into another equation.
- Elimination: Add or subtract equations to eliminate one variable and solve for the remaining variable(s).
- Gaussian elimination: A systematic process of row operations to transform the system into an upper triangular form and then back-substitute to find the solution.
- Matrix methods: Using matrices to represent the system and solve it using matrix operations.
Types of solutions:
- Unique solution: The system has exactly one set of values for the variables that satisfies all equations.
- Infinitely many solutions: The system has multiple solutions, often represented as a line or plane with parametric equations.
- No solution: The system is inconsistent, meaning there are no values for the variables that satisfy all equations.
Applications:
Systems of linear equations are used to model and solve real-world problems in various fields. Some examples include:
- Finding equilibrium points in economic models.
- Analyzing electrical circuits.
- Solving motion problems in physics.
- Designing computer graphics and animation.
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