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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.
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