I need five solutions to the above problem, but I'm not sure where to start with this.
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$\begingroup$The first step would be to eliminate the irritating 9. Then we have $3x \equiv 8-9 \equiv -1 \equiv 10 \pmod{11}$. Next we need to know how to divide by 3 by finding a number $a$ such that $3\cdot a \equiv 1 \pmod{11}$. You can work out the multiplication table mod 11 on your own and discover that $3\cdot 4=12 \equiv 1 \pmod{11}$ so $$4\cdot 3\cdot x \equiv 4\cdot 10 \pmod{11} \\ x \equiv 40 \equiv 7 \pmod{11}$$ Then we should just check that this solution is correct. $$3\cdot 7 + 9 \equiv 8 \pmod{11} \\ 21 + 9 \equiv 8 \pmod{11} \\ 30 \equiv 8\pmod{11}$$ And since $30=2\cdot 11 + 8 \equiv 0+8 \equiv 8 \pmod{11}$ our solution checks out.
$\endgroup$ $\begingroup$Definitions
Let Z be the Set of Integers.
Definition: If a ≡ c (mod b) then a = c+by where a,b,c,y ∈Z.
Given a set of equations, a variable in an equation is a basic variable if its coefficient is 1 and its coefficient in every other equation is 0. An equation with a basic variable is a basic equation. A system of equations where every equation is basic is a basic solution. The greatest common factor (GCF) of a set of integers is the largest integer that divides every element of the set.
By definition 3x+9≡8 (mod 11) is the same as the equation 3x+9=8+11y where x,y ∈Z.
Motivation
One way to find integer solutions that satisfy the equation (3x+9=8+11y ) would be to express each of the original variables 𝑥𝑥,𝑦 as a function of parameter variables (t0,t1,…), transform each original variable into a basic variable, assign integer values to the parameter variables in the basic solution, and compute the basic equations.
The transformation from an equation to a basic equation may be performed by algebraic manipulations. The transformation considered here judiciously adds variables and equations to the system, then considers the original equation plus the new equations as a system of linear equations, and solves the new system by back substitution. In every step if there is a GCF greater than 1 for all coefficients plus the right-hand constant in the current equation then the GCF should be removed before continuing.
With practice in performing the algebraic manipulations, an understanding of the motivation for expressing original variables in terms of parameter variables and transforming the equations into basic equations, it would be possible to organize the symbolic manipulations in terms of an augmented matrix and to manipulate the augmented matrix instead of manipulating the algebraic symbols. In either case the solution process has two phases: constraint and variable generation phase (CGP), and back substitution phase (BSP).
Solution
Reference
Chionglo, J. F. (2017). How I solved 3x + 9 == 8 (mod 11) by Symbolic and Augmented Matrix Manipulations. Available at .
$\endgroup$ $\begingroup$Another method: \begin{align} 3x+9&\equiv8\equiv-3&\pmod{11}\\ x+3&\equiv-1&\pmod{11}\\ x&\equiv-4\equiv7&\pmod{11} \end{align}
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