Solve the simultaneous equation, giving your answer in base 2.
11×+11y=1001
10×+y=11
× is the multiplication operator
x is a variable name
Don't you have an x key on your keyboard? (right next to z)
multiply the 2nd equation by 11 and you have
11x+11y = 1001
110x+11y = 1001
subtract to get
11x = 0
x = 0
so y = 11
To solve the simultaneous equations, we will use the method of substitution.
First, let's solve the second equation for y:
10x + y = 11
y = 11 - 10x
Now substitute this expression for y in the first equation:
11x + 11(11 - 10x) = 1001
Simplify the expression:
11x + 121 - 110x = 1001
Combine like terms:
-99x + 121 = 1001
Subtract 121 from both sides:
-99x = 880
Divide both sides by -99 to solve for x:
x = -880/-99
x = 8.888...
Now substitute this value of x back into the second equation to find y:
10(8.888...) + y = 11
88.888... + y = 11
y = 11 - 88.888...
y = - 77.888...
Converting these decimal values into binary:
x = 8.888... (in base 2) = 1000.111111...
y = -77.888... (in base 2) = 1111101.111111...
Therefore, the solutions to the simultaneous equations in base 2 are:
x = 1000.111111...
y = 1111101.111111...
To solve the simultaneous equations, we can use the method of substitution or elimination. Let's use the method of elimination:
Given equations:
1) 11x + 11y = 1001
2) 10x + y = 11
To eliminate one of the variables, we multiply equation 2 by 11:
(10x + y) * 11 = 11 * 11
110x + 11y = 121
Now we have two equations:
1) 11x + 11y = 1001
3) 110x + 11y = 121
Subtracting equation 1 from equation 3 eliminates the variable y:
(110x + 11y) - (11x + 11y) = 121 - 1001
99x = -880
Divide both sides by 99:
x = -880 / 99
x = -8.888...
To find the value of y, substitute the value of x into equation 2:
10(-8.888...) + y = 11
-88.888... + y = 11
Adding 88.888... to both sides:
y = 11 + 88.888...
y = 99.888...
Now let's convert the decimal answers to binary.
For x = -8.888..., it becomes -1000.111... in binary.
To convert the fractional part into binary:
0.111... * 2 = 1.222...
Since 1 is the integer part, we remove it and continue the process with the fractional part:
0.222... * 2 = 0.444...
0.444... * 2 = 0.888...
Continuing this process, we have:
0.111... (binary fraction) = 0.888... (decimal fraction)
Therefore, x = -1000.111... (binary).
For y = 99.888..., it becomes 1100011.111... in binary.
To convert the fractional part into binary:
0.111... * 2 = 1.222...
Since 1 is the integer part, we remove it and continue the process with the fractional part:
0.222... * 2 = 0.444...
0.444... * 2 = 0.888...
0.888... * 2 = 1.776...
Continuing this process, we have:
0.111... (binary fraction) = 1.776... (decimal fraction)
Therefore, y = 1100011.111... (binary).
So, the solution to the simultaneous equations in base 2 is:
x = -1000.111... (binary)
y = 1100011.111... (binary)