Solve the simultaneous equation 11 base x +10 base y = 5 base 10 and 13 base x + 7 base y = 19 base 10
converting to base 10
11 base x +10 base y = 5 base 10 ---> x+1 + y+0 = 5
x + y = 5
13 base x + 7 base y = 19 base 10 ---> x+3 + 7 = 19
x = 9
then in x+y=5 , we would have y = -4
Usually we would expect the "bases" to be whole numbers, but ....
check:
13x + 7y = 1910
and if x = 9 and y = -4
1(9)+3 + 7 = 19 , true
11x + 10y = 510
1(9)+1 + 1(-4)+0 = 5
9 - 4 = 5 , true
Cleared
To solve the simultaneous equations, we need to first convert the numbers provided in different bases to the same base. Let's convert the numbers to base 10.
Given:
1. 11x + 10y = 5 (in base 10)
2. 13x + 7y = 19 (in base 10)
To convert from base x/y to base 10, we need to expand each number using the respective digit powers of the base.
Let's convert the first equation:
11x + 10y = 5 (in base 10)
To convert the number 11 base x to base 10:
11 base x = (1 * x^1) + (1 * x^0) = x + 1
So the equation becomes:
(x + 1) + 10y = 5
Next, let's convert the second equation:
13x + 7y = 19 (in base 10)
To convert the number 13 base x to base 10:
13 base x = (1 * x^1) + (3 * x^0) = x + 3
So the equation becomes:
(x + 3) + 7y = 19
Now, we have a system of two equations that are in base 10:
1. (x + 1) + 10y = 5
2. (x + 3) + 7y = 19
We can now solve this system of equations using any suitable method such as substitution, elimination, or matrices to find the values of x and y.