11x+10y=10001
10x-y=10
Leave your answer in base 2
To solve this system of equations, let's multiply the second equation by 10 and add it to the first equation:
11x + 10y = 10001
10(10x - y) = 10(10)
100x - 10y = 100
Now we have the following equation:
111x = 10101
To find x, we divide both sides by 111:
x = 10101 / 111
Now let's convert x to base 2:
10101 (base 10) = 10000011110101 (base 2)
So, x = 10000011110101 (base 2).
To find y, we substitute the value of x back into the second equation:
10x - y = 10
10(10000011110101) - y = 10
100000111101010 - y = 10
- y = 10 - 100000111101010
y = 100000111101010 - 10
y = 100000111100000
So, y = 100000111100000 (base 2).
Therefore, the solution to the system of equations in base 2 is x = 10000011110101 and y = 100000111100000.
To solve these equations in base 2, we need to convert the numbers to binary form. Let's go step-by-step:
1. Convert 10001 to base 2:
10001 in base 10 is 10011100010001 in base 2.
2. Convert 10 to base 2:
10 in base 10 is 1010 in base 2.
3. Rewrite the equations in base 2:
11x + 10y = 10011100010001
10x - y = 1010
Now, we can solve this system of equations using a method called substitution or elimination. Let's use the elimination method:
4. Multiply the second equation by 10:
10(10x - y) = 10(1010)
which simplifies to:
100x - 10y = 10100
5. Subtract the second equation from the first equation:
11x + 10y - (100x - 10y) = 10011100010001 - 10100
which simplifies to:
11x + 10y - 100x + 10y = 10011001111101
Combine like terms:
-89x + 20y = 10011001111101
So, the simplified system of equations is:
-89x + 20y = 10011001111101
11x + 10y = 10011100010001
Unfortunately, since the equations involve variables in base 2, it is not possible to explicitly solve for x and y without numerical methods or converting back to base 10.
To solve this system of equations and express the answer in base 2, follow these steps:
Step 1: Rearrange the second equation to solve for y in terms of x:
10x - y = 10
y = 10x - 10
Step 2: Substitute the value of y from the rearranged equation into the first equation:
11x + 10(10x - 10) = 10001
11x + 100x - 100 = 10001
111x - 100 = 10001
111x = 10101
Step 3: Solve for x by dividing both sides by 111:
x = 10101 / 111
x = 91 in base 10
Step 4: Substitute the value of x back into the rearranged equation to solve for y:
y = 10(91) - 10
y = 910 - 10
y = 900 in base 10
Step 5: Convert the values of x and y to base 2:
x = 91 in base 10
91 = 2^6 + 2^4 + 2^3 + 2^0
91 = 1011011 in base 2
y = 900 in base 10
900 = 2^9 + 2^8 + 2^5 + 2^2
900 = 1110000100 in base 2
So the solution to the system of equations in base 2 is x = 1011011 and y = 1110000100.