No answer, or every answer

Two notes about one ticket counter can disagree. Let x be an adult ticket's price, in rupees. Let y be a student ticket's price. In the lab, note A is 2x + 3y = 340. Slide k in note B, 4x + 6y = k.

The rule

If elimination ends with a false sentence, like 0 = 30, the facts clash. They cannot both be true. There is no answer.

If elimination ends with 0 = 0, the two equations say one fact twice. The sentence 0 = 0 is true for every x and y. Every point of the one line is an answer.

Example

Note A says 2 adult and 3 student tickets cost ₹340. Note B says 4 adult and 6 student tickets cost ₹650. The first two ratios are 2/4 and 3/6. Both are 1/2.

The third ratio is 340/650. It is not 1/2. Double note A: 4x + 6y = 680. Subtract note B: 0 = 30. The two notes cannot both be true.

Change note B to 4x + 6y = 680. All three ratios are 1/2. Subtracting gives 0 = 0. Note B is note A doubled. Note B adds no new fact.

Try the lab to finish this lesson.