Solved on Feb 11, 2024

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

STEP 1

Assumptions
1. We are looking for integer solutions $(x, y)$ to the equation $3x + 7y = 2$ .
2. Integer solutions mean that both $x$ and $y$ 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 $x$ and $y$ that satisfy the equation.

STEP 3

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

STEP 4

Since $y$ must be an integer, the numerator $2 - 3x$ must be divisible by $7$ .

STEP 5

Let's find a value for $x$ such that $2 - 3x$ is divisible by $7$ . We can do this by checking the values of $x$ one by one until we find a suitable integer.

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

Using the formula from STEP_9, we can express the general solution for $x$ and $y$ :
$x = 3 + \frac{7}{1}t = 3 + 7t$ $y = -1 - \frac{3}{1}t = -1 - 3t$

STEP 12

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

STEP 13

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

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

STEP 1

Assumptions
1. We are looking for integer solutions $(x, y)$ to the equation $3x + 7y = 2$ .
2. Integer solutions mean that both $x$ and $y$ 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 $x$ and $y$ that satisfy the equation.

STEP 3

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

STEP 4

Since $y$ must be an integer, the numerator $2 - 3x$ must be divisible by $7$ .

STEP 5

Let's find a value for $x$ such that $2 - 3x$ is divisible by $7$ . We can do this by checking the values of $x$ one by one until we find a suitable integer.

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

STEP 10

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

STEP 11

Using the formula from STEP_9, we can express the general solution for $x$ and $y$ :
$x = 3 + \frac{7}{1}t = 3 + 7t$ $y = -1 - \frac{3}{1}t = -1 - 3t$

STEP 12

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

STEP 13

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

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

STEP 1

Assumptions1. We are looking for integer solutions (x,y)(x, y)(x,y) to the equation 3x+7y=23x + 7y = 23x+7y=2.2. Integer solutions mean that both xxx and yyy 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 xxx and yyy that satisfy the equation.

STEP 3

We can rewrite the equation in terms of yyy:y=2−3x7y = \frac{2 - 3x}{7}y=72−3x​

STEP 4

Since yyy must be an integer, the numerator 2−3x2 - 3x2−3x must be divisible by 777.

STEP 5

Let's find a value for xxx such that 2−3x2 - 3x2−3x is divisible by 777. We can do this by checking the values of xxx one by one until we find a suitable integer.

STEP 6

Upon checking, we find that x=3x = 3x=3 gives us 2−3(3)=2−9=−72 - 3(3) = 2 - 9 = -72−3(3)=2−9=−7, which is divisible by 777.

STEP 7

Substitute x=3x = 3x=3 into the equation to find the corresponding value of yyy:y=2−3(3)7=−77=−1y = \frac{2 - 3(3)}{7} = \frac{-7}{7} = -1y=72−3(3)​=7−7​=−1

STEP 8

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

STEP 9

STEP 10

First, we need to find gcd⁡(3,7)\gcd(3, 7)gcd(3,7). Since 333 and 777 are both prime numbers and do not share any common factors other than 111, we have:gcd⁡(3,7)=1\gcd(3, 7) = 1gcd(3,7)=1

STEP 11

Using the formula from STEP_9, we can express the general solution for xxx and yyy:x=3+71t=3+7tx = 3 + \frac{7}{1}t = 3 + 7tx=3+17​t=3+7t y=−1−31t=−1−3ty = -1 - \frac{3}{1}t = -1 - 3ty=−1−13​t=−1−3t

STEP 12

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

STEP 13

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

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

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

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 $x$ and $y$ that satisfy the equation.

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

Since $y$ must be an integer, the numerator $2 - 3x$ must be divisible by $7$ .

Let's find a value for $x$ such that $2 - 3x$ is divisible by $7$ . We can do this by checking the values of $x$ one by one until we find a suitable integer.

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

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

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

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

Using the formula from STEP_9, we can express the general solution for $x$ and $y$ :
$x = 3 + \frac{7}{1}t = 3 + 7t$ $y = -1 - \frac{3}{1}t = -1 - 3t$

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

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