Solve the simultaneous equations given your answers in base two 11x+11y=1001,10x-y=11

Answer 1

To solve the simultaneous equations given in binary (base two), we first need to convert them to decimal (base ten) to make solving easier, then convert the answers back to base two.

Given Equations in Binary

1. \( 11x + 11y = 1001 \) (in binary)
2. \( 10x - y = 11 \) (in binary)

Step 1: Convert to Decimal

1. \( 11 \) in binary is \( 3 \) in decimal.
2. \( 1001 \) in binary is \( 9 \) in decimal.
3. \( 10 \) in binary is \( 2 \) in decimal.
4. \( 11 \) in binary is \( 3 \) in decimal.

Now rewriting the equations in decimal:

1. \( 3x + 3y = 9 \)
2. \( 2x - y = 3 \)

Step 2: Solve the Equations

From the first equation, we can simplify:

\[ 3x + 3y = 9 \implies x + y = 3 \quad \text{(Divide everything by 3)} \] So, \[ y = 3 - x \quad \text{(Equation 3)} \]

Now substitute \( y \) from Equation 3 into the second equation:

\[ 2x - (3 - x) = 3 \]

Simplifying this:

\[ 2x - 3 + x = 3 \] \[ 3x - 3 = 3 \] \[ 3x = 6 \] \[ x = 2 \]

Now substitute \( x \) back into Equation 3 to find \( y \):

\[ y = 3 - 2 = 1 \]

Step 3: Convert the Solutions to Binary

Now we have:

\[ x = 2 \quad (\text{in decimal}) \implies x = 10 \quad (\text{in binary}) \] \[ y = 1 \quad (\text{in decimal}) \implies y = 1 \quad (\text{in binary}) \]

Final Answer

The solutions to the simultaneous equations in binary are:

\( x = 10 \) and \( y = 1 \) in base two.