Solve the simultaneous equationso

Question

11x-10y=101 base 2
10x+11y=1001 base 2

Answer 1

To solve this system of simultaneous equations, we can convert the numbers to base 10, solve the equations, and then convert the solution back to base 2.

The given equations are:

11x - 10y = 101 base 2 (equation 1)
10x + 11y = 1001 base 2 (equation 2)

To convert a number from base 2 to base 10, we multiply each digit by the corresponding power of 2 and sum them.

Let's convert the numbers in equation 1 from base 2 to base 10:
101 base 2 = (1 * 2^2) + (0 * 2^1) + (1 * 2^0) = 4 + 1 = 5
Therefore, equation 1 becomes:
11x - 10y = 5 (equation 3)

Now, let's convert the numbers in equation 2 from base 2 to base 10:
1001 base 2 = (1 * 2^3) + (0 * 2^2) + (0 * 2^1) + (1 * 2^0) = 8 + 1 = 9
Therefore, equation 2 becomes:
10x + 11y = 9 (equation 4)

Now we have the following system of equations in base 10:
11x - 10y = 5 (equation 3)
10x + 11y = 9 (equation 4)

We can solve this system by elimination or substitution. Let's use the elimination method.

Multiply equation 3 by 10 and equation 4 by 11 to eliminate the variable 'y':
110x - 100y = 50 (equation 5)
110x + 121y = 99 (equation 6)

Subtract equation 5 from equation 6 to eliminate 'x':
110x + 121y - (110x - 100y) = 99 - 50
221y + 100y = 49
321y = 49
y = 49 / 321

Now, substitute y = 49 / 321 into either equation 3 or 4 (let's use equation 3) to find x:
11x - 10(49 / 321) = 5
11x - 490 / 321 = 5
11x = 5 + 490 / 321
11x = 1605 / 321

Simplify the right-hand side:
11x = 5 + 5
11x = 10
x = 10 / 11

Therefore, the solution to the simultaneous equations is x = 10/11 and y = 49/321 in base 10.

To convert these numbers back to base 2, we can divide by 2 repeatedly and keep track of the remainders.

For x = 10 / 11:
Divide 10 by 2: 10 ÷ 2 = 5 remainder 0
Divide 5 by 2: 5 ÷ 2 = 2 remainder 1
Divide 2 by 2: 2 ÷ 2 = 1 remainder 0
Divide 1 by 2: 1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top, x = 1010 base 2.

For y = 49 / 321:
Divide 49 by 2: 49 ÷ 2 = 24 remainder 1
Divide 24 by 2: 24 ÷ 2 = 12 remainder 0
Divide 12 by 2: 12 ÷ 2 = 6 remainder 0
Divide 6 by 2: 6 ÷ 2 = 3 remainder 0
Divide 3 by 2: 3 ÷ 2 = 1 remainder 1
Divide 1 by 2: 1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top, y = 11001 base 2.

Therefore, the solution to the simultaneous equations in base 2 is x = 1010 and y = 11001.