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.
The two equations we are considering are:
1) 2x - 3y = 5
2) 4x - 6y = 10
To solve these equations, we can first notice that equation 2 is just equation 1 multiplied by 2. This means that the two equations are dependent on each other and represent the same line in the cartesian plane.
When two equations represent the same line, they have no point of intersection and thus no solution. This means that there is no value of x and y that satisfies both equations simultaneously.
To demonstrate this, let's plug in a few points into the original equations:
For example, let's try x = 1 and y = 1:
Equation 1: 2(1) - 3(1) = 5
Which simplifies to: 2 - 3 = 5
This is not true, so the point (1,1) does not satisfy equation 1.
Now let's try the same point in equation 2:
Equation 2: 4(1) - 6(1) = 10
Which simplifies to: 4 - 6 = 10
This is also not true, so the point (1,1) does not satisfy equation 2.
Since the point (1,1) does not satisfy either equation, there is no point at which both equations are true simultaneously. This confirms that the system of equations has no solution.