Can Someone Help Me Plz
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(I'd been curious myself how this could be explained in the most suggestive way.)
Example: 2x2 linear system
2*x - 3*y = 4
-2*x +4*y =6
It has an easy geometrical interpretation. Solve for y to make that clear:
y = 2/3 x - 4/3
y = 1/2 x + 3/2
These are two lines. Slopes are 2/3, 1/2 and y-intercepts are -4/3, 3/2 respectively.
**** Solving the 2x2 system means finding the point if intersection of both lines.****
(With 3x3 linear system: Find point of intersection of three planes in space, higher-rank systems: Find point of intersection of hyperplanes in multidimensional space, but don't panic - it's still just simple algebra).
1) Either equation obviously still holds if we multiply both sides by any real number.
2) What happens if we SUBSTRACT one equation from the other?:
y = 2/3 x - 4/3
y = 1/2 x + 3/2 --> 0 = (2/3 - 1/2) x - 4/3 - 3/2 --> 0 = 1/6 x - 17/6 --> x = 17.
The line "y = 2/3 x - 4/3" remains unchanged. The second line "y = 1/2 x + 3/2" has been mapped to a new, third line with the equation "x = 17", which is a parallel to the y-axis, crossing the x-axis at the point x=17 AND (!) also crossing the line "y = 2/3 x - 4/3" at the specific values of x and y that we had been looking for in the first place. Draw it and you'll find, that "y = 1/2 x + 3/2" has been rotated aound the point of intersection by a certain angle into "x = 17".
Since 1) states that we may multiply with anything, we can specifically multiply with "-1" and add equations instead of substracting them. Also, we need not necessarily "solve for y" as I did in the beginning of this example, but can right away start the business of multiplying with constant factor and adding/substracting. Both equations hold, whether they be given implicitly like "2*x - 3*y = 4" or in explicit form "y = 2/3 x - 4/3" - it doesn't matter.
Example: 2x2 linear system
2*x - 3*y = 4
-2*x +4*y =6
It has an easy geometrical interpretation. Solve for y to make that clear:
y = 2/3 x - 4/3
y = 1/2 x + 3/2
These are two lines. Slopes are 2/3, 1/2 and y-intercepts are -4/3, 3/2 respectively.
**** Solving the 2x2 system means finding the point if intersection of both lines.****
(With 3x3 linear system: Find point of intersection of three planes in space, higher-rank systems: Find point of intersection of hyperplanes in multidimensional space, but don't panic - it's still just simple algebra).
1) Either equation obviously still holds if we multiply both sides by any real number.
2) What happens if we SUBSTRACT one equation from the other?:
y = 2/3 x - 4/3
y = 1/2 x + 3/2 --> 0 = (2/3 - 1/2) x - 4/3 - 3/2 --> 0 = 1/6 x - 17/6 --> x = 17.
The line "y = 2/3 x - 4/3" remains unchanged. The second line "y = 1/2 x + 3/2" has been mapped to a new, third line with the equation "x = 17", which is a parallel to the y-axis, crossing the x-axis at the point x=17 AND (!) also crossing the line "y = 2/3 x - 4/3" at the specific values of x and y that we had been looking for in the first place. Draw it and you'll find, that "y = 1/2 x + 3/2" has been rotated aound the point of intersection by a certain angle into "x = 17".
Since 1) states that we may multiply with anything, we can specifically multiply with "-1" and add equations instead of substracting them. Also, we need not necessarily "solve for y" as I did in the beginning of this example, but can right away start the business of multiplying with constant factor and adding/substracting. Both equations hold, whether they be given implicitly like "2*x - 3*y = 4" or in explicit form "y = 2/3 x - 4/3" - it doesn't matter.
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