7x - 4y = 1

Definition

The equation 7x - 4y = 1 is a linear Diophantine equation, which is a type of equation that seeks integer solutions for the variables involved. In this case, it represents a straight line in the coordinate plane, where x and y are integers. The equation is essential for understanding relationships between integers and the conditions under which solutions exist.

5 Must Know Facts For Your Next Test

1. For the equation 7x - 4y = 1 to have integer solutions, the GCD of the coefficients (7 and -4) must divide the constant term (1). Since GCD(7, 4) = 1, there are integer solutions.
2. The general solution can be expressed as x = x0 + kt and y = y0 + lt, where (x0, y0) is a particular solution and k and l depend on the coefficients of x and y.
3. Using methods like the Extended Euclidean Algorithm helps find particular integer solutions to linear Diophantine equations.
4. The solutions form an infinite set since adding multiples of k and l will yield more integer pairs satisfying the equation.
5. Graphically, this equation represents a line in the xy-plane, where every point on that line corresponds to a pair of integers (x, y) that satisfy the equation.

Review Questions

• How can you determine whether the linear Diophantine equation 7x - 4y = 1 has integer solutions?
  • To determine if the linear Diophantine equation 7x - 4y = 1 has integer solutions, you first need to find the greatest common divisor (GCD) of the coefficients 7 and -4. The GCD is 1. Since this GCD divides the constant term (1), we conclude that there are indeed integer solutions for this equation.
• What steps would you take to find a particular solution for the equation 7x - 4y = 1 using the Extended Euclidean Algorithm?
  • To find a particular solution for the equation 7x - 4y = 1 using the Extended Euclidean Algorithm, start by applying the algorithm to express GCD(7, -4) as a linear combination of 7 and -4. This will involve a series of divisions and recording remainders until reaching zero. The last non-zero remainder will give you integer coefficients that can be rearranged to form a solution for x and y in terms of specific values.
• Evaluate how changing the constant term from 1 to another integer affects the existence of solutions for the equation 7x - 4y = k.
  • Changing the constant term in the equation from 1 to another integer k affects whether there are solutions based on whether k is divisible by the GCD of the coefficients. For instance, if k is an integer such that GCD(7, -4) divides k, then there will be integer solutions. If not, no integer solutions exist. Therefore, understanding this relationship is crucial when analyzing different values for k in linear Diophantine equations.

Related terms

Diophantine Equation: An equation that seeks integer solutions, typically involving two or more variables.

Integer Solutions: Values for the variables of an equation that are whole numbers, both positive and negative, including zero.

Greatest Common Divisor (GCD): The largest positive integer that divides each of the integers without leaving a remainder, important for determining the solvability of Diophantine equations.