(1/7)(1/2)B + (1/2)B + (1/4)B = (1/2)A - 12
and
(1/4)(1/2)A + (1/2)A = (1/2)B - 18
Simplifying these equations:
(1/14)B + (1/2)B + (1/4)B = (1/2)A - 12
Combining like terms, we get:
(7/28)B + (14/28)B + (7/28)B = (14/28)A - 24
Simplifying further:
(28/28)B = (14/28)A - 24
B = (1/2)A - 24
Substituting this into the second equation:
(1/8)A + (1/2)A = (1/2)((1/2)A - 24) - 18
Simplifying:
(5/8)A = (1/4)A - 30
Multiplying both sides by 8 to eliminate fractions:
5A = 2A - 240
3A = -240
A = -80
Substituting this into the equation for B:
B = (1/2)(-80) - 24 = -64
Therefore, the solution is A = -80 and B = -64.
AAAaannndd the bot gets it wrong yet again!
Simplified, the two equations are
14A - 23B = 336
5A - 4B = -144
so
A = -4656/59
B = -3696/59
I apologize for my mistake. Thank you for providing the correct solution.
To simplify the equations, let's start by finding a common denominator for the fractions. For the first equation:
(1/7)(1/2)B + (1/2)B + (1/4)B = (1/2)A - 12
The common denominator for the fractions is 4. So, multiply each fraction by 4:
(4/7)(1/2)B + (4/2)B + (4/4)B = (4/2)A - 48
(2/7)B + 2B + B = 2A - 48
Now, let's simplify the equation further. Combine like terms:
(2/7 + 1 + 1)B = 2A - 48
(10/7)B = 2A - 48
For the second equation:
(1/4)(1/2)A + (1/2)A = (1/2)B - 18
The common denominator for the fractions is 4. So, multiply each fraction by 4:
(4/4)(1/2)A + (4/2)A = (4/2)B - 72
(1/2)A + 2A = 2B - 72
Now, let's simplify the equation further. Combine like terms:
(1/2 + 2)A = 2B - 72
(5/2)A = 2B - 72
So, the simplified equations are:
(10/7)B = 2A - 48
(5/2)A = 2B - 72