The following simultaneous equations are given in base two
11X+10y=1100
101X-y=111
multiply the bottom by 10 and you have
0011x+10y = 1100
1010x-10y = 1110
Now add to get
1101x = 11010
x = 10
Now you can get y
To solve the given simultaneous equations, we will use the method of substitution.
Step 1: Rearrange the first equation to solve for X in terms of y.
11X + 10y = 1100
=> 11X = 1100 - 10y
=> X = (1100 - 10y)/11
Step 2: Substitute the value of X in the second equation.
101X - y = 111
=> 101((1100 - 10y)/11) - y = 111
=> (101 * (1100 - 10y) - 11y) = 111 * 11
=> (10100 - 1010y - 11y) = 1221
Step 3: Simplify the equation.
=> 10100 - 1010y - 11y = 1221
=> 10100 - 1021y = 1221
Step 4: Solve for y.
=> -1021y = 1221 - 10100
=> -1021y = -8887
=> y = -8887 / -1021
=> y = 8.7 (approx.)
Step 5: Substitute the value of y into the first equation to solve for X.
11X + 10(8.7) = 1100
=> 11X + 87 = 1100
=> 11X = 1100 - 87
=> 11X = 1013
=> X = 1013 / 11
=> X = 92 (approx.)
Thus, the approximate solutions to the simultaneous equations are X = 92 and y = 8.7
To solve these simultaneous equations in base two, we will use binary operations and convert the numbers into decimal form. Let's break it down step by step:
1. Convert the given binary numbers into decimal form:
- In the first equation, we have 1100 and 111. Converting them into decimal gives us:
1100(base 2) = 12(base 10)
111(base 2) = 7(base 10)
2. Rewrite the simultaneous equations in decimal form:
11X + 10y = 12
101X - y = 7
3. Solve the linear equations using any method you prefer, such as substitution or elimination. Let's use the substitution method:
From the second equation, we can write:
y = 101X - 7
Substitute this value of y into the first equation:
11X + 10(101X - 7) = 12
Simplify the equation:
11X + 1010X - 70 = 12
1021X - 70 = 12
Move terms around to isolate X:
1021X = 12 + 70
1021X = 82
Divide both sides by 1021:
X = 82 / 1021 ≈ 0.08
4. Substitute the value of X back into one of the equations to find the value of y. Let's use the second equation:
101X - y = 7
Substitute X = 0.08:
101(0.08) - y = 7
8.08 - y = 7
Solve for y by subtracting 8.08 from both sides:
-y = 7 - 8.08
-y = -1.08
Multiply both sides by -1 to isolate y:
y = 1.08
5. Convert the decimals back into binary form:
X ≈ 0.08(base 10) = 0.0001100110011001...(base 2)
y = 1.08(base 10) = 1.0001100110011...(base 2)
Therefore, the solutions to the simultaneous equations in base two are:
X ≈ 0.0001100110011001...(base 2)
y = 1.0001100110011...(base 2)