two systems of equations are given below for each system, choose the best description of it solution.
1) the system has no solution
2) the system has a unique solution
3) the system has a infinitity many solutions. they must satisfy the following equation
x+4y=8
-x-4y=-8
-x-2y=4
x+2y=4
If you add the first two equations, you will get 0 = 0
Any time you have a situation when the variables disappear and you end up with a true statement, there are an infinite number of solutions. Whatever (x,y) works in one equation will also work in the other.
The second set adds to 0 = 8
This is false which means there are no solutions. You cannot find an (x,y) that will work in both equations.
For the first system of equations:
1) x + 4y = 8
2) -x - 4y = -8
By adding the two equations together, we get:
(1) + (2) => (x - x) + (4y - 4y) = (8 - 8)
0 = 0
Since 0 = 0, this means that the two equations are equivalent and represent the same line. Therefore, the system has infinitely many solutions.
For the second system of equations:
1) -x - 2y = 4
2) x + 2y = 4
This system also has a unique solution. By adding the two equations together, we get:
(1) + (2) => (-x + x) + (-2y + 2y) = (4 + 4)
0 = 8
Since 0 does not equal 8, this means that there is no solution to this system of equations.
To determine the best description for each system's solution, we need to examine the number of solutions they have. Here's how to do it:
1) For the first system:
The equations are:
x + 4y = 8
-x - 4y = -8
Add the two equations together:
(x + 4y) + (-x - 4y) = 8 + (-8)
0 = 0
From this, we can see that both variables cancel out, resulting in 0 = 0. This means that the equations are equivalent and represent the same line. When two equations represent the same line, they have infinitely many solutions. Therefore, the best description for this system is "the system has infinitely many solutions."
2) For the second system:
The equations are:
-x - 2y = 4
x + 2y = 4
Add the two equations together:
(-x - 2y) + (x + 2y) = 4 + 4
0 = 8
In this case, adding the equations results in a contradiction. 0 does not equal 8. This means the equations have no common solution, and the lines they represent are parallel. Therefore, the best description for this system is "the system has no solution."
In summary:
1) The first system has infinitely many solutions.
2) The second system has no solution.