Solving the 1/a + 1/b + 1/c = 3 Equation

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In summary, the conversation discusses solving for a,b,c given the equations 1/a + 1/b + 1/c = 3 and ⁴√(a³) + ⁴√(b³) + ⁴√(c³) ≥ ³√(a²) + ³√(b²) + ³√(c²). The solution is found to be a = b = c = 1 or a,b,c > 1/3. The conversation also mentions the possibility of solving for whole numbers a,b,c.
  • #1
c6_viyen_1995
10
0
a,b,c >0

1/a + 1/b + 1/c = 3

How can solve that
⁴√(a³) + ⁴√(b³) + ⁴√(c³) ≥ ³√(a²) + ³√(b²) + ³√(c²)

thank you very much!
 
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  • #2
Help me!
 
  • #3
Well, I can see that one solution is a=b=c=1 (you probably already know that, though :wink:).

Past that, I don't know. One number must be greater than one, one less, and one relates to the other two by the second formula.
 
  • #4
c6_viyen_1995 said:
a,b,c >0

1/a + 1/b + 1/c = 3

How can solve that
⁴√(a³) + ⁴√(b³) + ⁴√(c³) ≥ ³√(a²) + ³√(b²) + ³√(c²)

thank you very much!

If you start with 1/a=3, you find a = 1/3.
With 1/a + 1/b = 3, you can see that a > 1/3, because b > 0.
Due to the symmetry of a and b, that means also that b > 1/3.

With 1/a + 1/b + 1/c = 3, it follows that a,b,c > 1/3.

I suspect that is the intended answer.

Otherwise, if a,b,c are supposed to be whole numbers, it follows that a,b,c ≥ 1.
It follows that then a = b = c = 1, because otherwise we would have fractions.


The second equation can be "solved" equivalently. Do you see how?
 
  • #5


I would approach solving this equation by first understanding the mathematical principles involved. The equation 1/a + 1/b + 1/c = 3 is a type of rational equation, which means the variables (a, b, and c) are in the denominator of a fraction.

To solve this equation, we can use algebraic techniques such as finding a common denominator and simplifying the fractions. We can also use the concept of reciprocals, where the reciprocal of a number is its inverse (e.g. the reciprocal of 2 is 1/2).

Once we have a solution for a, b, and c in the first equation, we can then substitute those values into the second equation and solve for the inequality. In this case, we can use the properties of exponents to simplify the expressions and compare the values.

Overall, solving these equations requires a strong understanding of algebraic principles and the ability to manipulate equations to find a solution. It is important to note that any solution found must also be checked to ensure it satisfies the given conditions (a, b, and c must be greater than 0).
 

What is the 1/a + 1/b + 1/c = 3 equation and why is it important?

The 1/a + 1/b + 1/c = 3 equation is a mathematical expression that represents a relationship between three numbers. It is important because it is a common form of the equation used in many areas of science and engineering, such as physics, chemistry, and electrical engineering.

What is the process for solving the 1/a + 1/b + 1/c = 3 equation?

The process for solving this equation involves finding the values of a, b, and c that satisfy the equation. This can be done by rearranging the equation to isolate one variable, then substituting in values for the other variables and solving for the remaining variable. This process may need to be repeated multiple times to find all possible solutions.

Are there any special cases or restrictions for solving the 1/a + 1/b + 1/c = 3 equation?

Yes, there are some special cases and restrictions to keep in mind when solving this equation. First, all variables must be nonzero real numbers. Additionally, any combination of positive and negative numbers can be used for a, b, and c, as long as they satisfy the equation. However, there are certain combinations of numbers that will not have a real solution, such as when all three variables are negative or when one variable is zero.

What are some real-life applications of the 1/a + 1/b + 1/c = 3 equation?

This equation has many real-life applications, particularly in fields that involve rates, proportions, or relationships between multiple quantities. For example, it can be used in chemistry to calculate the rate of a chemical reaction, in physics to determine the resistance of a circuit, or in biology to study the relationship between different species in an ecosystem.

Are there any tips or strategies for solving the 1/a + 1/b + 1/c = 3 equation more efficiently?

Yes, there are a few tips and strategies that can help make solving this equation easier. One approach is to simplify the equation by finding common factors or using algebraic techniques, such as multiplying both sides by a common denominator. Another tip is to carefully choose values for one or two variables to make the equation easier to solve. Additionally, checking for extraneous solutions and double-checking your work can help avoid mistakes and save time in the long run.

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