Solve the given equation or indicate that there is no soluti

Solve the given equation or indicate that there is no solution. (If there is no solution, enter NO SOLUTION.) 4x + 2 = 3 in Z_6 x = in Z_6

Solution

Given :

4x + 2 = 3

subtract 2 from both sides, we get:

4x+2-2 = 3-2

4x = 1 .......................(1)

x = 1/4

Now look for the inverse of the number 4 (we use the word inverse here and elsewhere to mean multiplicative inverse because we have the word negative to use for the additive inverse of a number). In other words, look for a number we can multiply by 4 to get the multiplicative identity 1. there isn’t one. The reason is that the row of 4 in the times table of Z6 does not have 1. So 4 has no inverse.

Hence, the equation 4x = 1 has no solutions.

Since 4x= 1 is equivalent to our original equation(1), it follows that :

4x + 2 = 3 has no solution.