11x+10y=10001
10x-y=10
Solve the equations leaving your answer in base 2
To solve the equations, we can use the method of substitution.
First, let's isolate y in the second equation:
10x - y = 10
y = 10x - 10
Now, substitute this expression for y in the first equation:
11x + 10y = 10001
11x + 10(10x - 10) = 10001
11x + 100x - 100 = 10001
111x - 100 = 10001
111x = 10101
x = 10101/111
x = 91 in base 10
Now, substitute this value of x back into the second equation to find y:
10x - y = 10
10(91) - y = 10
910 - y = 10
y = 900
So, the solution in base 10 is x = 91 and y = 900.
Now, let's convert the solution to base 2.
To convert x to base 2, divide it by 2 repeatedly and take the remainders. Start from the bottom and read up to get the binary representation.
91 / 2 = 45 remainder 1
45 / 2 = 22 remainder 0
22 / 2 = 11 remainder 0
11 / 2 = 5 remainder 1
5 / 2 = 2 remainder 1
2 / 2 = 1 remainder 0
1 / 2 = 0 remainder 1
Reading the remainders from bottom to top, we have x = 1011011 in base 2.
To convert y to base 2, we perform the same division process:
900 / 2 = 450 remainder 0
450 / 2 = 225 remainder 0
225 / 2 = 112 remainder 1
112 / 2 = 56 remainder 0
56 / 2 = 28 remainder 0
28 / 2 = 14 remainder 0
14 / 2 = 7 remainder 0
7 / 2 = 3 remainder 1
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1
Reading the remainders from bottom to top, we have y = 1110010100 in base 2.
Therefore, the solution in base 2 is x = 1011011 and y = 1110010100.
To solve the given system of equations, we'll use the method of substitution.
First, let's solve the second equation for y:
10x - y = 10
Adding y to both sides:
10x = y + 10
Subtracting 10 from both sides:
y = 10x - 10
Now, substitute this expression for y in the first equation:
11x + 10(10x - 10) = 10001
11x + 100x - 100 = 10001
Combine like terms:
111x - 100 = 10001
Add 100 to both sides:
111x = 10101
Divide both sides by 111:
x = 10101 / 111
Now, convert x to base 2:
10101 divided by 111 in base 2 is 91 in base 10.
Thus, the solution for x in base 10 is 91, which is equivalent to 1011011 in base 2.
Next, substitute the value of x back into either of the two equations to find y:
y = 10x - 10
y = 10(91) - 10
y = 910 - 10
y = 900
Therefore, the solution to the given system of equations in base 2 is:
x = 1011011
y = 111000100
To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of elimination:
Given equations:
1) 11x + 10y = 10001
2) 10x - y = 10
First, we need to eliminate one of the variables from the equations by manipulating the equations. We'll eliminate the "y" variable by multiplying equation 2 by 10 and adding it to equation 1. This will eliminate "y" and allow us to solve for "x":
Multiply equation 2 by 10:
10 * (10x - y) = 10 * 10
100x - 10y = 100
Now, add equation 2 (multiplied by 10) to equation 1:
(11x + 10y) + (100x - 10y) = 10001 + 100
11x + 100x = 10101
111x = 10101
Divide both sides of the equation by 111 to solve for "x":
x = 10101 / 111
x = 91
Now, substitute the value of "x" back into equation 2 and solve for "y":
10(91) - y = 10
910 - y = 10
-y = 10 - 910
-y = -900
Divide both sides of the equation by -1 to solve for "y":
y = -900 / (-1)
y = 900
Therefore, the solution to the system of equations is x = 91 and y = 900.
Now, to convert the answer to base 2, we can use a method called binary conversion.
To convert the decimal number to binary form, divide the decimal number repeatedly by 2 until the quotient becomes 0. The remainders at each division will give us the binary representation of the number.
Converting x = 91 to binary:
91 divided by 2 gives a quotient of 45 and a remainder of 1 (LSB).
45 divided by 2 gives a quotient of 22 and a remainder of 0.
22 divided by 2 gives a quotient of 11 and a remainder of 0.
11 divided by 2 gives a quotient of 5 and a remainder of 1.
5 divided by 2 gives a quotient of 2 and a remainder of 1.
2 divided by 2 gives a quotient of 1 and a remainder of 0.
1 divided by 2 gives a quotient of 0 and a remainder of 1 (MSB).
Therefore, 91 in decimal is equal to 1011011 in binary.
Converting y = 900 to binary:
900 divided by 2 gives a quotient of 450 and a remainder of 0.
450 divided by 2 gives a quotient of 225 and a remainder of 0.
225 divided by 2 gives a quotient of 112 and a remainder of 1.
112 divided by 2 gives a quotient of 56 and a remainder of 0.
56 divided by 2 gives a quotient of 28 and a remainder of 0.
28 divided by 2 gives a quotient of 14 and a remainder of 0.
14 divided by 2 gives a quotient of 7 and a remainder of 0.
7 divided by 2 gives a quotient of 3 and a remainder of 1.
3 divided by 2 gives a quotient of 1 and a remainder of 1.
1 divided by 2 gives a quotient of 0 and a remainder of 1.
Therefore, 900 in decimal is equal to 1110000100 in binary.
Thus, the solution in base 2 is x = 1011011 and y = 1110000100.