College algebra. Need help! thanks..

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  • Explain the method for finding the solution of a system of linear equations using row operations. 
Jun 25th, 2015

Thank you for the opportunity to help you with your question!

For example, 5x + 5y - 3z = 16; 2x - 5y +10z = 20; 4x + 2y +7z = 9

Row operations means to take two of the equations at a time and eliminate one of the variables.  For instance, the first and second equations you can add together since the "y" variable would be eliminated (+5y - 5y = 0y). Similarly the 2nd and 3rd qeuations could be used to eliminate the "x" variable as follows:

2x - 5y + 10z = 20

4x + 2y + 7z = 9

Multiply the top equation by -2 then add them together

-4x +10y -20z = -40 (this is the 2x - 5y + 10z = 20 equation multiplied by 2 on both sides)

4x + 26y + 7z = 9

0x + 36y - 13z = -31 (solution to adding top and bottom equations together)

By changing the equations and using them to eliminate one variable each time, you eventually end up with the x- y- and z-coordinates that fit all 3 equations (in other words, the intersection point of all 3 lines).  Sometimes not all three lines intersect.  Two of them may intersect or none of them may intersect.

This method is a precursor to solving by matrices (linear algebra).  Linear algebra is more advanced but working with matrices is similar to what we are doing here.

Row operations includes multiplying or dividing a row (or 2 rows) by numbers so that a variable will be "cancelled out".

Such as 3x + 2y = 14

and 2x + 4y = 20

Multiply to top row by 2 and the bottom by -3 so you have:

6x + 4y = 24

-6x - 12y = -60

Now you have 0x - 8y = -36

-8y = -36

8y = 36

y = 36/8

Please let me know if you need any clarification. I'm always happy to answer your questions.
Jun 25th, 2015

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