11x+10y=10001

Question

10x-y=10
Solve the equations leaving your answer in base 2

Answer 1

To solve the equations, we can use the method of substitution.

First, let's isolate y in the second equation:

10x - y = 10
y = 10x - 10

Now, substitute this expression for y in the first equation:

11x + 10y = 10001
11x + 10(10x - 10) = 10001
11x + 100x - 100 = 10001
111x - 100 = 10001
111x = 10101
x = 10101/111
x = 91 in base 10

Now, substitute this value of x back into the second equation to find y:

10x - y = 10
10(91) - y = 10
910 - y = 10
y = 900

So, the solution in base 10 is x = 91 and y = 900.

Now, let's convert the solution to base 2.

To convert x to base 2, divide it by 2 repeatedly and take the remainders. Start from the bottom and read up to get the binary representation.

91 / 2 = 45 remainder 1
45 / 2 = 22 remainder 0
22 / 2 = 11 remainder 0
11 / 2 = 5 remainder 1
5 / 2 = 2 remainder 1
2 / 2 = 1 remainder 0
1 / 2 = 0 remainder 1

Reading the remainders from bottom to top, we have x = 1011011 in base 2.

To convert y to base 2, we perform the same division process:

900 / 2 = 450 remainder 0
450 / 2 = 225 remainder 0
225 / 2 = 112 remainder 1
112 / 2 = 56 remainder 0
56 / 2 = 28 remainder 0
28 / 2 = 14 remainder 0
14 / 2 = 7 remainder 0
7 / 2 = 3 remainder 1
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1

Reading the remainders from bottom to top, we have y = 1110010100 in base 2.

Therefore, the solution in base 2 is x = 1011011 and y = 1110010100.