The following simultaneous equations are given in base two 11x + 10y = 1100
101x - y =111
Find the value of x and y
multiply the 2nd by 10
1010x - 10y = 1110
add to the first
1010x - 10y = 1110
11x + 10y = 1100
...............................
1101x = 11010
x = 10
in the 1st
110 +10y = 1100
10y = 1100-110 = 110
y = 11
I DON'T UNDERSTAND
When I calculated1010+11 I got something else
Well, I must say, solving equations in base two is quite a special kind of arithmetic circus! Let me put on my clown shoes and dive into it!
Alrighty, let's start solving your simultaneous equations:
11x + 10y = 1100 -- (Equation 1)
101x - y = 111 -- (Equation 2)
To make things easier, let's convert these base-two numbers into good old decimal!
In Equation 1, let's convert 1100 (base two) to decimal to get:
11x + 10y = 12
Now, in Equation 2, we need to convert 101 (base two) to decimal and 111 (base two) to decimal. Doing that, we get:
101x - y = 5
Alright! So now we have a nice little number problem to solve. Let's proceed:
Let's isolate y in Equation 1:
10y = 12 - 11x
y = (12 - 11x)/10
Now, let's substitute this value of y into Equation 2:
101x - [(12 - 11x)/10] = 5
We can simplify this equation by multiplying both sides by 10 to get rid of that pesky fraction:
1010x - (12 - 11x) = 50
Distributing the negative sign:
1010x - 12 + 11x = 50
Combining like terms:
1021x - 12 = 50
To isolate x, let's add 12 to both sides:
1021x = 62
Finally, divide both sides by 1021:
x = 62/1021
Now that we have the value of x, we can substitute it back into Equation 1 to find y. But, alas, we shall need a calculator with base-two capabilities to convert the final answer back into binary form.
So, my friend, with calculations in base two turning into arithmetic acrobatics, I'm afraid I can't provide you with an exact answer for x and y. You'll need to grab a calculator that supports base-two calculations to complete the final step.
Have a jolly time solving those equations, and remember, math can be more fun than a circus act!
To solve the given simultaneous equations, we will use the method of substitution. The steps involved are as follows:
1. Convert the given base two numbers to their decimal equivalents:
- 1100 in base 2 = 1*(2^3) + 1*(2^2) = 12 in decimal
- 101 in base 2 = 1*(2^2) + 0*(2^1) + 1*(2^0) = 5 in decimal
- 111 in base 2 = 1*(2^2) + 1*(2^1) + 1*(2^0) = 7 in decimal
2. Rewrite the equations using the decimal numbers:
- 11x + 10y = 12
- 101x - y = 7
3. Solve one of the equations for one variable in terms of the other variable:
Let's solve the second equation for x:
- 101x - y = 7
- Rearrange the equation to isolate x:
101x = y + 7
x = (y + 7) / 101
4. Substitute the expression for x obtained in step 3 into the other equation:
- 11x + 10y = 12
- Substitute (y + 7) / 101 for x:
11*((y + 7) / 101) + 10y = 12
5. Solve the equation obtained in step 4 for y:
- Multiply through by 101 to eliminate the fraction:
11*(y + 7) + 1010y = 1212
- Distribute and simplify:
11y + 77 + 1010y = 1212
1021y + 77 = 1212
1021y = 1135
y = 1135 / 1021
6. Substitute the value of y obtained in step 5 back into the expression for x from step 3, and calculate x:
- x = (y + 7) / 101
x = (1135 / 1021 + 7) / 101
Finally, calculate the values of x and y using the obtained expressions.