2x + 3y = 5

Definition

The equation 2x + 3y = 5 represents a linear Diophantine equation, which is a type of equation where the solutions are required to be integers. This equation describes a line in a two-dimensional coordinate system, where any integer point (x, y) that satisfies the equation corresponds to a valid solution. Understanding this equation is crucial for finding integer solutions and analyzing the conditions under which they exist.

5 Must Know Facts For Your Next Test

1. To determine if the linear Diophantine equation 2x + 3y = 5 has integer solutions, one can check if the greatest common divisor (gcd) of the coefficients (2 and 3) divides the constant term (5), which it does since gcd(2, 3) = 1.
2. The general solution to the equation can be expressed as x = x_0 + 3t and y = y_0 - 2t, where (x_0, y_0) is a particular solution and t is any integer.
3. The line represented by the equation 2x + 3y = 5 can be graphed on a Cartesian plane, revealing all pairs of (x, y) that satisfy the equation.
4. Finding all integer solutions involves exploring how values of x and y change while ensuring that both remain integers.
5. The intercepts of the line can be found by setting x or y to zero; for this equation, when x = 0, y = rac{5}{3}, and when y = 0, x = rac{5}{2}.

Review Questions

• How can you determine if the equation 2x + 3y = 5 has integer solutions?
  • To determine if the equation has integer solutions, you check if the greatest common divisor (gcd) of the coefficients of x and y divides the constant term. In this case, gcd(2, 3) is 1, which divides 5. Therefore, we can conclude that there are integer solutions to this linear Diophantine equation.
• Describe how to find a particular solution for the equation 2x + 3y = 5 and explain how it relates to the general solution.
  • To find a particular solution for the equation, you can choose specific integer values for x or y. For example, if you let x = 1, then you can solve for y to find that y = 1. The general solution is derived from this particular solution by expressing x and y in terms of an integer parameter t. This means any solution can be generated by adjusting t while maintaining integer values for both x and y.
• Analyze how changing the coefficients in the linear Diophantine equation affects its solvability and integer solutions.
  • Changing the coefficients in the linear Diophantine equation directly impacts its solvability. If you increase or decrease the coefficients while still maintaining their relationship with the constant term, it can lead to different gcd values. If the gcd no longer divides the constant term, then there will be no integer solutions. For example, modifying the coefficients could create scenarios where certain values of t yield non-integer results for either x or y, therefore altering the set of possible solutions significantly.

Related terms

Linear Equation: An equation that graphs to a straight line, typically represented in the form ax + by = c, where a, b, and c are constants.

Diophantine Equation:

A polynomial equation that seeks integer solutions, named after the ancient Greek mathematician Diophantus.

Integer Solutions: Values of variables in an equation that are whole numbers, which can be positive, negative, or zero.