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solotion algebretically

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2 Answers

You know that we can Multiply the Numerator and denominator of a fraction by the same number and the value of the fraction does not change:
 
    1/2 = (2 * 1) / ( 2 * 2) = 3/ 6 = 4/ 8 =  ....... 50/100
 
     This says that if we split a unit, first into 2 part, 2nd into 4 part, 3rd into 6 part, ...... , and into 100 part
 
     1 of the first split, 2 of the 4 split, 3 of the 6 split, ..... , 50 of the 100 split
 
       will all add up the same quantity.
 
      Now:
                   if we have  1/2 = x / 6
 
                    We have 2 fraction equal, therefore, 6 divided by 2 = 3 , X has to be multiplied by 3 to keep                     the values of 2 fractions equal
                     to solve this equation:
                       1/2 = x/6
 
                       We multiply both sides by 6
 
                        6*1/2  = 6 *X / 6
                           
                        Now divide both sides by 6
 
                          Simplify each fraction
                           
                             X = 3
                So , that is where the cross multiplication of 2 equal fraction comes into picture.
                  Multiply both fraction with LCM of the denominators, and then simplify each fraction to find the
                   value of the variable.  
 
 
Hello again Sherry -
The first equation above is called an algebraic rational expression. This is just a fraction with variables in it. Look at the equation. If there were no x's in it, what would you do? Isn't it just a proportion? It is two fractions set equal to each other. I would start solving this by cross multiplying. Then you get the following: 
 
-2(x-2) = (4-x)(x-2) 
 
Now distribute the negative 2 on the left and FOIL the right side, combine like terms and get this: 
 
x-8x +12 = 0
 
Now can you factor the quadratic? When I do this I get
 
(x-6)(x-2) 
 
Now use the Zero Product property, set each factor equal to zero and solve. You get two answers, 6 and 2. BUT..... you need to check you answers. What happens in the denominator of the original equation if x = 2? Uh-oh! the denominator would be equal to zero, so 2 is called an "excluded value", and the only valid solution is 6. 
 
Let's check; 4-6/6-2 = -2(6-2)? 
-2/4 = -2(4) 
Voila! it checks!