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1/(a+b) + 1/c = 4/(a+b+c)

3[(a+2b/(2c-a)]-(2c+a+b)/c=

User R Reveley
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1 Answer

1 vote

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)

User Marcos Santana
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