Consider the two equations you put a square around, which form a system with no solution. Solve the equations. What do you notice about these equations? What does it mean to have no solution? Plug a few points into the original equations and check to show that they can have no solution. Explain your work.
Let's consider the two equations:
1) 2x + 4y = 10
2) 2x + 4y = 5
To solve these equations, let's first subtract one equation from the other:
(2x + 4y) - (2x + 4y) = 10 - 5
0 = 5
This is a contradiction, showing that these equations have no solution.
When a system of equations has no solution, it means that the two lines represented by the equations are parallel and never intersect. This occurs when the slopes of the two lines are equal but the y-intercepts are different.
To show that these equations have no solution, we can plug in some points into the original equations and see that they do not satisfy both equations at the same time.
For example, let's take the point (1,1):
1) 2(1) + 4(1) = 10
2 + 4 = 10
6 does not equal 10
1) 2(1) + 4(1) = 5
2 + 4 = 5
6 does not equal 5
Therefore, the point (1,1) does not satisfy both equations simultaneously, proving that these equations have no solution.