Solving the Equation 3xy 7 and Its Interpretation

Solving Linear Equations: The given equation, 3x y 7, is a linear equation in two variables. To find the value of y in terms of x, one can isolate y on one side of the equation.

Step-by-Step Solution: Isolating y

Starting with the original equation:

3x y 7

Subtracting 3x from both sides, we get:

y 7 - 3x

This equation now expresses y in terms of x. For each value of x, there will be a corresponding value for y. Let’s explore a few examples:

For x 0:

y 7 - 3(0) 7

So the solution is (0, 7).

For x 1:

y 7 - 3(1) 4

So another solution is (1, 4).

For x 2:

y 7 - 3(2) 1

So yet another solution is (2, 1).

These points lie on a straight line, as the equation represents a linear relationship between x and y.

Algebraic Rearrangement

The equation can also be rearranged into the slope/y-intercept form (y mx c), which is y -3x 7. This form is useful for graphing and understanding the relationship between x and y more clearly. Let’s consider a few more points:

For x 0:

y -3(0) 7 7

So the point is (0, 7).

For x 1:

y -3(1) 7 4

So the point is (1, 4).

For x 2:

y -3(2) 7 1

So the point is (2, 1).

Exploring Diophantine Equations

Since the problem statement was a bit ambiguous, we can consider the equation as a linear Diophantine equation and solve it. The GCD of 3 and 7 is 1, which allows us to use the extended Euclidean algorithm to find a particular solution:

Using the Extended Euclidean Algorithm

Let's begin with the Euclidean algorithm to find the coefficients:

1. 7 3(2) 1

2. 3 3(1) 0

So, 7 - 3(2) 1.

Using the extended Euclidean algorithm, we can find the general solution for x and y:

x 2 - 3k

y 1 7k

where ( k in mathbb{Z} ).

To find specific integer solutions, let's pick ( k 0, 1, -1 ):

For k 0:

x 2 - 3(0) 2

y 1 7(0) 1

For k 1:

x 2 - 3(1) -1

y 1 7(1) 8

For k -1:

x 2 - 3(-1) 5

y 1 7(-1) -6

These solutions all satisfy the equation 3x y 7.

Conclusion

The equation 3x y 7 has an infinite number of solutions if no additional constraints are provided. However, using the solutions derived from the Euclidean algorithm, we can express the general solutions as (2 - 3k, 1 7k) for all integers k. For practical purposes, the slope/y-intercept form and specific values can be used to find solutions within a certain range.