Let's simplify the given expression step by step:
Given expression:
1. (1/(a+b)) + (1/c) = 4/(a+b+c)
Step 1:
To combine the fractions on the left side, find a common denominator, which is (a+b)c:
[(c + a + b)/(c(a+b))] = 4/(a+b+c)
Step 2:
Cross-multiply:
(c + a + b)(a+b+c) = 4c(a+b)
Step 3:
Expand both sides of the equation:
a(a + b + c) + b(a + b + c) + c(a + b + c) = 4ac + 4bc
Step 4:
Rearrange the terms:
a^2 + ab + ac + ab + b^2 + bc + ac + bc + c^2 = 4ac + 4bc
Step 5:
Combine like terms:
a^2 + 2ab + 2ac + b^2 + 2bc + c^2 = 4ac + 4bc
Step 6:
Move all terms to one side of the equation:
a^2 + 2ab + 2ac - 4ac + b^2 + 2bc - 4bc + c^2 = 0
Step 7:
Simplify further:
a^2 + 2ab - 2ac + b^2 - 2bc + c^2 = 0
Now let's simplify the second expression:
Given expression:
2. 3[(a + 2b)/(2c - a)] - (2c + a + b)/c
Step 1:
Multiply the entire expression by c to eliminate the fraction in the denominator:
3c[(a + 2b)/(2c - a)] - (2c + a + b)
Step 2:
Distribute 3c to the terms in the numerator:
(3ac + 6bc)/(2c - a) - (2c + a + b)
Step 3:
Find a common denominator for the first term:
[(3ac + 6bc) - (2c + a + b)(2c - a)]/(2c - a)
Step 4:
Expand the second term in the numerator:
[(3ac + 6bc) - (4c^2 - a^2 - 2ac + ab)]/(2c - a)
Step 5:
Combine like terms in the numerator:
(3ac + 6bc - 4c^2 + a^2 + 2ac - ab)/(2c - a)
Step 6:
Simplify the expression:
(a^2 + 5ac - ab + 6bc - 4c^2)/(2c - a)
So, the simplified expressions are:
1. a^2 + 2ab - 2ac + b^2 - 2bc + c^2 = 0
2. (a^2 + 5ac - ab + 6bc - 4c^2)/(2c - a)