Thread: system of liner equation

Hello everyone. I have a question about system of liner equation.
In wiki,I found an statement which is talking about it is possible for a system of two equations and two unknowns to have no solution (if the two lines are parallel), or for a system of three equations and two unknowns to be solvable (if the three lines intersect at a single point).
I understand that the parallel lines have no common intersect point .So there exist no solution. However, I don’t understand the last one. This question defeats me for a long time. Can someone help me please?

Originally Posted by kenpoon

In wiki,I found an statement which is talking about it is possible for a system of two equations and two unknowns to have no solution (if the two lines are parallel), or for a system of three equations and two unknowns to be solvable (if the three lines intersect at a single point).

I don't think you have finished explaining the gap in your understanding.

I take two ideas here.

A system of 3 eqns each with 2 unknowns have one (or maybe infinite) solution(s) if they intersect at the same point.

A system of 3 eqns each with 2 unknowns have no solutions if they all share the same gradient but intersect the y-axis at different points.

Thank you for your reply. I don't understand that the 3 eqns each with 2 unknowns have infinite solution(s) if they intersect at the same point.
Why they have infinite solutions? They are just intersect at one point only.

Think about the three equations 2x+ 3y= 4, x- y= 5, 4x- 4y= 20 in the two unknowns, x and y. The last two equations are "dependent" (4x- 4y= 20 is just 4(x- y= 5) so they are not really "two" equations) so this really the same as two equations in two unknowns. That is the case where three equations in two unknowns have a single solution.

Think about the three equations 2x+ 3y= 4, 6x+ 9y= 12, 8x+ 12y= 16 in the two unknowns, x and y. Those are all really the same equation (the second is 3 times the first and the third is 4 times the first) so any x and y that satisfy 2x+ 2y= 4 satisfy all three. Of course they do NOT "itersect at the same point". I believe pickslides misspoke there.