How do you find the integer solution of a linear equation?
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How do you find the integer solution of a linear equation?
Let a,b∈Z with a≠0.
- If a divides b, then the equation ax=b has exactly one solution that is an integer.
- If a does not divide b, then the equation ax=b has no solution that is an integer.
Who Solved the Diophantine equation?
Named in honour of the 3rd-century Greek mathematician Diophantus of Alexandria, these equations were first systematically solved by Hindu mathematicians beginning with Aryabhata (c. 476–550).
How many solutions does a linear Diophantine equation have?
General Solution to Linear Diophantine Equations. In the example above, an initial solution was found to a linear Diophantine equation. This is just one solution of the equation, however. When integer solutions exist to an equation a x + b y = n , ax+by=n, ax+by=n, there exist infinitely many solutions.
What is the linear Diophantine equation?
A Linear Diophantine equation (LDE) is an equation with 2 or more integer unknowns and the integer unknowns are each to at most degree of 1. Linear Diophantine equation in two variables takes the form of ax+by=c, where x,y∈Z and a, b, c are integer constants. x and y are unknown variables.
What are the integer solutions?
Integer solutions to an equation means solutions which are in the set of integers (see first line of my answer). The problem (x+3)(x-1) = 0 has two integer solutions, namely x=-3 and x=1. Try plugging them into the equation and you will see they make it true. So, since they are integers they are integer solutions.
What is linear Diophantine?
A Linear Diophantine equation (LDE) is an equation with 2 or more integer unknowns and the integer unknowns are each to at most degree of 1. Linear Diophantine equation in two variables takes the form of ax+by=c, where x,y∈Z and a, b, c are integer constants.
What is a integer solution?
An integer solution is a solution such that all the unknowns take integer values). Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations.
What is a linear Diophantine equation?
A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents. Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations.
What does it mean to solve a linear Diophantine equation?
Solving a linear Diophantine equation means that you need to find solutions for the variables x and y that are integers only. Finding integral solutions is more difficult than a standard solution and requires an ordered pattern of steps.
What is the difference between integer and Diophantine problems?
An integer solution is a solution such that all the unknowns take integer values). Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations.
How do you find all integer solutions of a homogeneous equation?
Substituting x 0 and y 0 to the solution of a homogeneous equation, we obtain all integer solutions of the given equation: