Part 1: COMPOSING. We need to get a common denominator for all three equations by multiplying the bottom of each denominator by the other denominators. After doing this, don't forget to multiply the numberators by it as well! After you get the numerators, combine like terms and put it over the common denominator (which does not need to be FOILED out). That will be your composed equation!
Part 2: To start decomposing, set your equation to A/x+5 + B/x+2 +C/x-7. Do the same thing you did to get common denominators and also for your numerators. (Notice you'll have letters in your equation now!) Set it equal to the numerator we found in the previous part. Combine like terms again and cross off the x^2 and x's to make it a lot easier!
Part 3: Plug in the coefficients into your matrix. After you have that plugged in, find the rref. Your resulting answer should be as shown above. Notice that A=4, B=2, and C=3! It lines up with the equations from the two earlier steps!
We have successfully decomposed the equation back to its original equation after we composed it! :- )
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