Equations with fractions: 2nd Level Let us now compare equations with fractions with adding fractions. To add these fractions --
-- their LCM is abc.
We have multiplied each numerator -- 2, 3, 4 -- by those factors of abc that are missing from its denominator. Now, when we clear this equation -- which has the same fractions:
-- each term in the equation is the same as the numerator in the addition. Each numerator -- 2, 3, 4 -- is multiplied by those factors of the LCM that are missing from its denominator. The terms of the cleared equation are the same as the numerators Example 1. Solve for x:
Again, when solving an equation with fractions, the very next statement you write -- the next line -- should have no fractions. Problem 9. Solve for x:
The LCM is the product of the three denominators. Here is the cleared equation and its solution:
The original equation is immediately transformed into an equation without fractions. Each succesive statement -- each line -- follows from the previous line. The transformations are a logical sequence of statements, as in Lesson 9. Problem 10. Solve for x:
Problem 11. Factor the denominators, clear of fractions, and solve for x:
The LCM is x(x − 2)(3x + 1). Here is the cleared equation and its solution:
Problem 12. Factor the denominators, clear of fractions, and solve for x:
The LCM is (x + 3)(x − 3)(x − 1). Here is the cleared equation and its solution:
This is a simple fractional equation, which we saw in Lesson 9. It is
reciprocal.
Whatever multiplies on one side will divide on the other. And whatever divides on one side will multiply on the other.
We would like x to be in the numerator on the left. Imagine placing it there.
reciprocal
Problem 13. Solve for x:
Problem 14. Solve for x:
In each of the following, solve for x.
Example 4. Solving for the reciprocal.
While we could solve this by clearing of fractions, there is the more
We have
This implies
Therefore, on taking reciprocals
For if two numbers are equal, then their reciprocals are also equal.(Except, if the number is 0.) In each of the following, solve for x by first solving for its reciprocal.
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