Name: Jason

Who is asking: Student
Level: Secondary

Question:
The linear systems with the following pattern all have the same solution. Is their some kind of reasoning as to why this is so?

2x+3y=4
3x+4y=5

4x+5y=6
5x+6y=7

The solution is always x=-1, y=2

Thank you
Jason

Hi Jason

In any of the pairs of equations that you have if you subtract the first from the second you get

x + y = 1 or x = 1 - y.

All of these equations have the pattern

ax + (a+1)y = a + 2

so substituting x = 1 - y into this equation gives

a(1-y) + (a+1)y = a+2
or
a - ay + ay + y = a + 2

Hence y = 2 and since x + y = 1, x = -1.

Cheers
Penny.