# Find the solutions of ## 7x+3y\equiv 6\pmod {11} ##.

• Math100
In summary, using the concept of inverses in modular arithmetic, we can find the solutions of the given system of congruences: ##x\equiv 9\pmod {11}## and ##y\equiv 3\pmod {11}##.
Math100
Homework Statement
Find the solutions of the following system of congruences:
## 7x+3y\equiv 6\pmod {11} ##
## 4x+2y\equiv 9\pmod {11} ##.
Relevant Equations
None.
Consider the following system of congruences:
## 7x+3y\equiv 6\pmod {11} ##
## 4x+2y\equiv 9\pmod {11} ##.
Then ## gcd(7\cdot 2-3\cdot 4, 11)=gcd(2, 11)=1 ##.
This means that ## \exists ## a unique solution.
Observe that
\begin{align*}
&2(7x+3y)-3(4x+2y)\equiv [2(6)-3(9)]\pmod {11}\\
&\implies 2x\equiv -15\pmod {11}\\
&\implies 2x\equiv 7\pmod {11}\\
&\implies 12x\equiv 42\pmod {11}\\
&\implies x\equiv 9\pmod {11}.\\
\end{align*}
Thus
\begin{align*}
&4(7x+3y)-7(4x+2y)]\equiv [4(6)-7(9)]\pmod {11}\implies -2y\equiv -39\pmod {11}\\
&\implies 2y\equiv 39\pmod {11}\implies 12y\equiv 234\pmod {11}\\
&\implies y\equiv 3\pmod {11}.\\
\end{align*}
Therefore, the solutions are ## x\equiv 9\pmod {11} ## and ## y\equiv 3\pmod {11} ##.

Math100 said:
Homework Statement:: Find the solutions of the following system of congruences:
## 7x+3y\equiv 6\pmod {11} ##
## 4x+2y\equiv 9\pmod {11} ##.
Relevant Equations:: None.

Consider the following system of congruences:
## 7x+3y\equiv 6\pmod {11} ##
## 4x+2y\equiv 9\pmod {11} ##.
Then ## gcd(7\cdot 2-3\cdot 4, 11)=gcd(2, 11)=1 ##.
This means that ## \exists ## a unique solution.
Observe that
\begin{align*}
&2(7x+3y)-3(4x+2y)\equiv [2(6)-3(9)]\pmod {11}\\
&\implies 2x\equiv -15\pmod {11}\\
&\implies 2x\equiv 7\pmod {11}\\
&\implies 12x\equiv 42\pmod {11}\\
&\implies x\equiv 9\pmod {11}.\\
\end{align*}
Thus
\begin{align*}
&4(7x+3y)-7(4x+2y)]\equiv [4(6)-7(9)]\pmod {11}\implies -2y\equiv -39\pmod {11}\\
&\implies 2y\equiv 39\pmod {11}\implies 12y\equiv 234\pmod {11}\\
&\implies y\equiv 3\pmod {11}.\\
\end{align*}
Therefore, the solutions are ## x\equiv 9\pmod {11} ## and ## y\equiv 3\pmod {11} ##.
Or you could observe that:
##\displaystyle 7x+3y+4x+2y\equiv 6+9 \pmod {11} ##

so that ##\displaystyle 11x+5y\equiv 15 \pmod {11} ##

which reduces to ##\displaystyle 0x+5y\equiv 4 \pmod {11} ## .

After a bit of work ##\displaystyle y\equiv 3 \pmod {11} ## .

Math100 said:
Homework Statement:: Find the solutions of the following system of congruences:
## 7x+3y\equiv 6\pmod {11} ##
## 4x+2y\equiv 9\pmod {11} ##.
Relevant Equations:: None.

Consider the following system of congruences:
## 7x+3y\equiv 6\pmod {11} ##
## 4x+2y\equiv 9\pmod {11} ##.
Then ## gcd(7\cdot 2-3\cdot 4, 11)=gcd(2, 11)=1 ##.
This means that ## \exists ## a unique solution.
Observe that
\begin{align*}
&2(7x+3y)-3(4x+2y)\equiv [2(6)-3(9)]\pmod {11}\\
&\implies 2x\equiv -15\pmod {11}\\
&\implies 2x\equiv 7\pmod {11}\\
&\implies 12x\equiv 42\pmod {11}\\
&\implies x\equiv 9\pmod {11}.\\
\end{align*}
Thus
\begin{align*}
&4(7x+3y)-7(4x+2y)]\equiv [4(6)-7(9)]\pmod {11}\implies -2y\equiv -39\pmod {11}\\
&\implies 2y\equiv 39\pmod {11}\implies 12y\equiv 234\pmod {11}\\
&\implies y\equiv 3\pmod {11}.\\
\end{align*}
Therefore, the solutions are ## x\equiv 9\pmod {11} ## and ## y\equiv 3\pmod {11} ##.
Correct.

I have observed that the coefficients of ##x## add up to ##11## so I simply added both equations and got ##5y\equiv 4\pmod{11}.## I got from ##5\cdot 9 \equiv 4\cdot 3\equiv 1\pmod{11}## the inverses ##5^{-1}\equiv 9\pmod{11}## and ##4^{-1}\equiv 3\pmod{11}.## Thus ##y\equiv 4\cdot 9\equiv 3\pmod{11} ## and ##x\equiv 4^{-1} (9-2y)\equiv 3\cdot (9-2\cdot 3)\equiv 9\pmod{11}.##

Last edited:
SammyS and Math100

## 1. What is the meaning of "solutions" in this equation?

In this equation, "solutions" refer to the values of x and y that satisfy the given congruence. These values will make the equation true when substituted into the equation.

## 2. How do you solve this congruence equation?

To solve this equation, we can use the Chinese Remainder Theorem or the Euclidean algorithm. We can also use trial and error by plugging in different values for x and y until we find a solution that satisfies the congruence.

## 3. Can this equation have multiple solutions?

Yes, this equation can have multiple solutions. In fact, there are an infinite number of solutions for this congruence equation.

## 4. What is the significance of the modulus (11) in this equation?

The modulus (11) in this equation represents the number that x and y are congruent to. In other words, the solutions for x and y must leave a remainder of 6 when divided by 11.

## 5. Can this equation be solved for any values of x and y?

No, this equation can only be solved for values of x and y that are congruent to 6 modulo 11. If the values of x and y do not satisfy this congruence, then there will be no solutions for the equation.

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