Consider the [ 2x +3 = 2x + 7 , 4x + 22 = 8x +10 5x + 5 = 5x +4 ] 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 solutions?
Plug a few points into the original equations and check to show that they can have no
solution. Explain your work
Let's first solve the equations:
1. 2x + 3 = 2x + 7
Subtract 2x from both sides:
3 = 7
This is a contradiction, so there is no solution to this equation.
2. 4x + 22 = 8x + 10
Subtract 4x from both sides:
22 = 4x + 10
Subtract 10 from both sides:
12 = 4x
Divide by 4:
x = 3
But when we plug x = 3 back into the equation, it doesn't hold true. So, there is no solution to this equation.
3. 5x + 5 = 5x + 4
Subtract 5x from both sides:
5 = 4
This is a contradiction, so there is no solution to this equation.
In all three equations, we ended up with a false statement (like 3 = 7 or 5 = 4). This means that the system of equations is inconsistent and has no solution.
To demonstrate this, let's plug in a few points:
1. For equation 1: Let x = 0
2(0) + 3 = 2(0) + 7
3 = 7
This is false, so x = 0 is not a solution.
2. For equation 2: Let x = 1
4(1) + 22 = 8(1) + 10
26 = 18
This is false, so x = 1 is not a solution.
3. For equation 3: Let x = 2
5(2) + 5 = 5(2) + 4
15 = 14
This is false, so x = 2 is not a solution.
In conclusion, the system of equations has no solution because the equations are inconsistent and have no common values that satisfy all of them simultaneously.