Question

An integer solution to the equation 3x+4=7y is an ordered pair of integers (x, y) that satisfies the equation. For example, (1,1) is one such solution. Write the set of all integer solutions to the equation 3x + 4 = 7y in set builder notation.

Answer 1

Answer on Question #74549 – Math – Discrete Mathematics

Question

An integer solution to the equation $3x + 4 = 7y$ is an ordered pair of integers $(x, y)$ that satisfies the equation. For example, $(1, 1)$ is one such solution. Write the set of all integer solutions to the equation $3x + 4 = 7y$ in set builder notation.

Solution

$$3x + 4 = 7y$$ $$3x = 7y - 4$$

Residue method

The left and the right side of the equation are divided into 3 groups.

if $y = 3m, m \in \mathbb{Z}$ then $7y - 4 = 7(3m) - 4 = 21m - 4 \quad (21m - 4) \mod 3 \neq 0$

if $y = 3m + 1, m \in \mathbb{Z}$ then $7y - 4 = 7(3m + 1) - 4 = 21m + 3 \quad (21m + 3) \mod 3 = 0$

if $y = 3m + 2, m \in \mathbb{Z}$ then $7y - 4 = 7(3m + 2) - 4 = 21m + 10 \quad (21m + 10) \mod 3 \neq 0$

$$y = 3m + 1, \quad m \in \mathbb{Z}$$ $$3x = 7y - 4 = 7(3m + 1) - 4 = 21m + 3$$ $$x = 7m + 1$$

Answer:

$$\{(x; y) \mid x = 1 + 7m, y = 1 + 3m, m \in \mathbb{Z}\}$$

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