|
Question 590116: Tell how many solutions the system has. do not actually solve.
5x+4y=-1
25x+20y=-5
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You are given the system of two equations:
.
5x+4y=-1
25x+20y=-5
.
and are asked how many solutions this system has. Just the way this question is asked might suggest that there could be something unusual involved.
.
You can tell the number of solutions by understanding how many points in common the graphs of the two equations have in common. There are three possibilities as follows:
.
(1) The linear (straight line) graphs for each equation have different slopes and therefore, these graphs cross at only one point. The coordinate pair for that point is the only solution to such a system. or
.
(2) The linear (straight line) graphs for each equation have the same slope and this presents two possibilities. First, either the graphs are separate parallel lines (like railroad tracks) and they never cross. Therefore there are no common solutions. And second, it could be possible that the two graphs lie on top of each other, so that every possible solution for one equation is also a solution for the other. In such a case there are an infinite number of common solutions for the two equations.
.
This problem is an example of the last possibility. The two equations have an infinite number of common solutions. How can you tell? Look again at the two equations:
.
5x+4y=-1
25x+20y=-5
.
If you multiply the top equation (all terms on both sides) by 5 you do not change the equation. However, when you do the multiplication by 5 the top equation becomes:
.
25x + 20y = -5
.
Notice that this is identical to the bottom equation. Therefore, the two equations have the same graphs. That means that the graph of the solution points for the top equation lies on top of the graph of the solution points for the bottom equation. This tells you that the graphs have an infinite number of common solution points which translates to an infinite number of common solutions.
.
I hope that this helps you to understand the three possible solutions that a system of two linear equations can have. And how you can picture the graphs of each equation to help you understand the number of points that will be solutions for a particular system of linear equations.
.
|
|
|
| |
