Question

Tell wheather the system has one solution, infinitely many solutions, or no solution.

1. 5x-3y=10
10x+6y=20

2. 2x-5y=17
6x-15y=51

Answer 1

1. One solution: (x, y) = (2, 0)
2. Infinitely many solutions: 2x -5y = 17

Explanation:

1. When you put the second equation in standard form, it is ...

... 5x +3y = 10

This is a different equation than the first one, so there will be one solution where the lines interect. (Adding the two equations gives 10x=20 ⇒ x=2. Since adding or subtracting y gives the same result, y must be zero.)

2. When you put the second equation in standard form, it is the same as the first:

... 2x -5y = 17 . . . . . divide the equation by 3

That is, any (x, y) values that satisfy the first equation will also satisfy the second equation (since they are the same). There are an infinite number of (x, y) values that do so.