Answer:
4, 1, 2, 3 . . . . . the systems of equations corresponding to the given solutions, in order
Step-by-step explanation:
When there are this many to do, it is convenient to let a calculator or spreadsheet help. Most will work best with integer coefficients, so the pairs of equations can be written ...
1.
- x - y = 25
- 2x + 3y = 180 . . . solution (51, 26)
2.
- 2x - 3y = -5
- 11x + y = 550 . . . solution (47, 33)
3.
- x - y = 19
- -12x + y = 168 . . . solution (-17, -36)
4. Here, we have a problem. The given equations don't match any solutions. The only remaining solution is (-31, -56), and it gives a value of -6 for the first equation, not 6. The second equation needs parentheses to make it work. We think your problem is ...
... 2x -y = -6, (x -305)/3 = 2y
Recast the way the others are above, these equations become ...
- 2x - y = -6
- x - 6y = 305 . . . solution (-31, -56)
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Comment on solution to systems of equations
The solution to ...
is ...
- x = (bf -ec)/(bd -ea)
- y = (cd -a·f)/(bd -ea) . . . . . (same denominator as for x)
This is a modified form of Cramer's rule. It is modified in that both numerator and denominator have the terms swapped (are the negative of the terms in the classic Cramer's rule). This form can be easy to remember, as the pattern is fairly simple. (Look at where the coefficients are in the equations. Use your spatial memory to remember the pattern.)
Example:
Applying these solution equations to system (4), we have ...
... x = ((-1)·305 -(-6)(-6))/((-1)·1 -(-6)·2) = -341/11 = -31
... y = ((-6)·1 -305·2) = -616/11 = -56
