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How Do You Solve a Rational Equation?

Solve the equation 1/f = 1/a + 1/b for f.

Summary

  1. Our contains three variables: 'f', 'a', and 'b'
  2. We want to solve for the 'f', which is in the denominator of the first fraction
  3. Since each denominator is unique, the LCD is the product of all three: 'abf'
  4. Multiplying each term by the LCD will help us eliminate any fractions in our equation
  5. abf(1/f) = ab(1) = ab
  6. abf(1/a) = bf(1) = bf
  7. abf(1/b) = af(1) = af
  8. We can factor an 'f' out of 'bf+af' to get: 'f(b+a)'
  9. Dividing by (a+b), or (b+a), will give us 'f' by itself on the right
  10. So we get: 'f = (ab)/(a+b)'

Notes

    1. Our equation is '1/f = 1/a + 1/b'
    2. 'f', 'a', and 'b' are all variables
    1. The variables in our equation are: 'f', 'a', and 'b'
    2. Our problem says to solve for 'f'
    1. Our problem says to solve for 'f'
    2. Our first fraction is '1/f', where 'f' is the variable we want to solve for
    1. Our problem says to solve for 'f', so we want to get it by itself
    2. Our equation is '1/f = 1/a + 1/b'
    1. Our equation is '1/f = 1/a + 1/b'
    2. 'f' is the variable we want to solve for
    1. Our first fraction is '1/f', where 'f' is the variable we want to solve for
    2. We need to figure out how to get 'f' out of the denominator and by itself
    1. The fractions in our equation are '1/f', '1/a', and '1/b'
    2. So we'll need to look at the denominators of each fraction
    1. The fractions in our equation are '1/f', '1/a', and '1/b'
    2. The denominators of each fraction are 'f', 'a', and 'b'
    1. The denominators of each fraction are 'f', 'a', and 'b'
    2. Since each denominator is unique, the LCD is the product of all three: 'abf'
    1. Since each denominator is unique, the LCD is the product of all three: 'abf'
    1. We determined the LCD to be 'abf'
    2. Multiplying 'abf' by each term will help us get 'f' by itself
    1. Left of the equal sign, we have abf(1/f) = ab(1) = ab
    1. Right of the equal sign, we have abf(1/a + 1/b) = abf(1/a) + abf(1/b) = bf(1) + af(1) = bf + af
    1. Our equation now looks like: 'ab = bf + af'
    1. Up to this point, our equation looked like: 'ab = bf + af'
    2. On the right side, both terms contain 'f' so we factor an 'f' out to get: 'f(b+a)'
    1. Up to this point, our equation looked like: 'ab = f(b + a)'
    2. Recognize that dividing by '(a+b)' is the same as dividing by '(b+a)
    3. Remember, if we divide one side by '(a+b)', we must divide the other side by the same
    1. The division by '(a+b)' gave us 'f' by itself!
    2. Remember, 'f' is the variable we were trying to solve for