Solving Rational Number Equation | Express x as Ratio of Integers

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In summary, based on the given equations, it can be concluded that x is rational and can be expressed as a ratio of two integers. To find the ratio, the equations can be rewritten in a different form and solved.
  • #1
sjaguar13
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Suppose a, b, and c are integers and x, y, and z are nonzero real numbers that satisfy the following equations:
xy/x+y = a
xz/x+z = b
yz/y+z = c

Is x rational? If so, express it as a ratio of two integers.



I am pretty sure x is rational, but I don't know how to get the ratio. I am guessing the ratio uses a, b, or c. I tried solving for x, but that got me no where.
 
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  • #2
Now, I have to assume that you mean xy/(x+y), with the brackets. You really should write those brackets in instead of leaving it to those trying to help you having to guess what you mean (technically, without the brackets, the equations become 2y = a, 2z = b = c, and x could be any non-zero, so we have to guess you mean it with the brackets). Now:

[tex]x = \frac{ya}{y - a},\ y = \frac{zc}{z - c},\ z = \frac{xb}{x - b}[/tex]

[tex]x = \frac{ya}{y - a},\ y = \frac{\left (\frac{xb}{x - b} \right )c}{\left (\frac{xb}{x - b} \right ) - c}[/tex]

Take your equation for x, substitute what you have for y, and you have an equation with only x, a, b, and c. Isolate x, and see what you get. I would think you'd get something like [itex]f(a,b,c)/g(a,b,c)[/itex] where the functions f and g are simple ones that take a, b, and c and simply add or multiply or divide them together in different ways. If this is so, and you get a non-zero denominator and numerator, then x is rational.
 
  • #3
sjaguar13 said:
Suppose a, b, and c are integers and x, y, and z are nonzero real numbers that satisfy the following equations:
xy/x+y = a
xz/x+z = b
yz/y+z = c

Is x rational? If so, express it as a ratio of two integers.



I am pretty sure x is rational, but I don't know how to get the ratio. I am guessing the ratio uses a, b, or c. I tried solving for x, but that got me no where.
These equations can be re-written as
[tex]{1\over x}+{1\over y}={1\over a}[/tex]
etc. In this form, they are very easy to solve.
 
  • #4
Krab is right provided that you define [tex] x^{-1} =u [/tex] and the like for y and z and work with those auxiliary variables
 

1. What is a rational number equation?

A rational number equation is an equation that involves fractions or decimals, where the numerator and denominator are both integers. These equations can be solved by finding the value of the variable that makes the equation true.

2. How do I express x as a ratio of integers?

To express x as a ratio of integers, you need to rewrite the equation so that the variable x is on one side and the constant is on the other side. Then, you can simplify the fraction by dividing both the numerator and denominator by their greatest common factor.

3. Can I solve a rational number equation without converting to fractions?

Yes, you can solve a rational number equation without converting to fractions. One way to do this is by multiplying both sides of the equation by the common denominator. This will eliminate the fractions and leave you with a simpler equation to solve.

4. What is the importance of solving rational number equations?

Solving rational number equations is important because it allows us to find the value of the variable in a problem and make sense of real-world situations. These equations are often used in fields such as science, engineering, and finance.

5. Can rational number equations have more than one solution?

Yes, it is possible for a rational number equation to have more than one solution. This means that there are multiple values of the variable that can make the equation true. It is important to check your solutions to make sure they work in the original equation.

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