What falls out when nothing is left
Elimination removes one letter from a pair of equations. What is left gives the verdict. The verdict is one solution, none, or a whole line.
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2x+3y=12 and x−y=1.- Eliminate x from
2x+3y=12andx−y=1. 5y = 10falls out. A number: one solution.- Eliminate x from
2x+3y=12and2x+3y=18. 0 = −6falls out. False: no solution.- Eliminate x from
2x+3y=12and4x+6y=24. 0 = 0falls out. True and empty: a whole line.
The rule
Rule: What elimination leaves gives the verdict. The leftover has a letter, or is false, or is true.
A leftover with a letter, like 5y = 10, is the one-crossing case. Here 2x + 3y = 12 minus 2x − 2y = 2 leaves 5y = 10. The second is x − y = 1 doubled. Solve it. Then put the value in an equation to find the other letter. A false leftover with no letter, like 0 = −6, means no solution. A true leftover with no letter, like 0 = 0, means one line. Both equations draw it.
| Leftover | Kind | Verdict |
|---|---|---|
5y = 10 | has a letter | exactly one pair |
0 = −6 | false, empty | no pair |
0 = 0 | true, empty | every point of one line |
Now take 2x + 3y = 12 with 4x + 6y = 25. Scale the first by 2: 4x + 6y = 24. Subtract it from 4x + 6y = 25. The left side is 0. The right side is 1. So the leftover is 0 = 1. It is false, so the pair is inconsistent. The lines run parallel.
Watch out!
"0 = 0 means any pair works" is wrong. Only the points of that one line work. The pair (100, 100) fails 2x + 3y = 12, because 200 + 300 is not 12.