Solved on Feb 11, 2024

Find all integer solutions to the linear equation 3x+7y=23x + 7y = 2.

STEP 1

Assumptions
1. We are looking for integer solutions (x,y)(x, y) to the equation 3x+7y=23x + 7y = 2.
2. Integer solutions mean that both xx and yy must be integers.

STEP 2

First, we will try to find a particular solution to the equation. To do this, we can use the method of trial and error to find values of xx and yy that satisfy the equation.

STEP 3

We can rewrite the equation in terms of yy:
y=23x7y = \frac{2 - 3x}{7}

STEP 4

Since yy must be an integer, the numerator 23x2 - 3x must be divisible by 77.

STEP 5

Let's find a value for xx such that 23x2 - 3x is divisible by 77. We can do this by checking the values of xx one by one until we find a suitable integer.

STEP 6

Upon checking, we find that x=3x = 3 gives us 23(3)=29=72 - 3(3) = 2 - 9 = -7, which is divisible by 77.

STEP 7

Substitute x=3x = 3 into the equation to find the corresponding value of yy:
y=23(3)7=77=1y = \frac{2 - 3(3)}{7} = \frac{-7}{7} = -1

STEP 8

We have found a particular solution to the equation: (x,y)=(3,1)(x, y) = (3, -1).

STEP 9

Now we need to find all integer solutions. To do this, we will use the fact that if (x0,y0)(x_0, y_0) is a particular solution to the equation ax+by=cax + by = c, then all solutions can be written in the form:
x=x0+bgcd(a,b)tx = x_0 + \frac{b}{\gcd(a, b)}t y=y0agcd(a,b)ty = y_0 - \frac{a}{\gcd(a, b)}t
where tt is an integer and gcd(a,b)\gcd(a, b) is the greatest common divisor of aa and bb.

STEP 10

First, we need to find gcd(3,7)\gcd(3, 7). Since 33 and 77 are both prime numbers and do not share any common factors other than 11, we have:
gcd(3,7)=1\gcd(3, 7) = 1

STEP 11

Using the formula from STEP_9, we can express the general solution for xx and yy:
x=3+71t=3+7tx = 3 + \frac{7}{1}t = 3 + 7t y=131t=13ty = -1 - \frac{3}{1}t = -1 - 3t

STEP 12

Since tt is an integer, we can substitute any integer value for tt to find all the integer solutions to the equation.

STEP 13

The set of all integer solutions (x,y)(x, y) to the equation 3x+7y=23x + 7y = 2 is given by:
x=3+7tx = 3 + 7t y=13ty = -1 - 3t
where tt is any integer.

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