Solve the simultaneous equation given your answer in base two 11x +11y= 1001. 10x-y = 11
X=11 while Y=110
To solve the simultaneous equations:
1) 11x + 11y = 1001
2) 10x - y = 11
Let's rearrange equation 2) to express y in terms of x:
2) -y = 11 - 10x
y = 10x - 11
Now, substitute this value of y into equation 1):
11x + 11(10x - 11) = 1001
11x + 110x - 121 = 1001
121x = 1122
x = 1122 / 121
x ≈ 9.28
Now, substitute the value of x back into equation 2) to solve for y:
y = 10x - 11
y = 10(9.28) - 11
y ≈ 92.8 - 11
y ≈ 81.8
Therefore, the solution to the simultaneous equations in base two is x ≈ 1001.01 and y ≈ 1010001.
To solve the given simultaneous equation and express the answer in base two, you can follow these steps:
Step 1: Rewrite the equations in standard form:
11x + 11y = 1001 (Equation 1)
10x - y = 11 (Equation 2)
Step 2: Rearrange Equation 2 to solve for x:
10x - y = 11
10x = y + 11
x = (y + 11) / 10 (Equation 3)
Step 3: Substitute Equation 3 into Equation 1:
11((y + 11) / 10) + 11y = 1001
Simplify this equation by multiplying both sides by 10:
11(y + 11) + 110y = 10010
11y + 121 + 110y = 10010
121y + 121 = 10010
121y = 9889
Step 4: Solve for y:
y = 9889 / 121
y ≈ 81.74
Step 5: Substitute the value of y into Equation 3 to find x:
x = (81.74 + 11) / 10
x ≈ 9.47
Step 6: Convert the decimal solutions, x ≈ 9.47 and y ≈ 81.74, into base two.
To convert decimal numbers to binary numbers, first convert the integer part and then the fractional part.
For x ≈ 9.47:
- The integer part of 9 is 1001 in base two.
- The decimal part of 0.47 can be converted by multiplying it by 2 repeatedly.
0.47 * 2 = 0.94 → 0
0.94 * 2 = 1.88 → 1
0.88 * 2 = 1.76 → 1
...
The fractional part converges to 0.01111... in base two.
Therefore, x ≈ 1001.01111 in base two.
For y ≈ 81.74:
- The integer part of 81 is 1010001 in base two.
- The decimal part of 0.74 can be converted as follows:
0.74 * 2 = 1.48 → 1
0.48 * 2 = 0.96 → 0
0.96 * 2 = 1.92 → 1
0.92 * 2 = 1.84 → 1
...
The fractional part converges to 0.10111... in base two.
Therefore, y ≈ 1010001.10111 in base two.
Hence, the simultaneous equation's solution in base two is:
x ≈ 1001.01111
y ≈ 1010001.10111
11x + 11y = 1001
divide by 11 and you have
x + y = 11
10x-y = 11
now add to get
11x = 110
x = 10
so y = 1