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.
Let's solve the equations:
1. 2x + 3 = 2x + 7
Subtracting 2x from both sides:
3 = 7
This equation has no solution.
2. 4x + 22 = 8x + 10
Subtracting 4x from both sides:
22 = 4x + 10
Subtracting 10 from both sides:
12 = 4x
Dividing by 4:
x = 3
This equation has a solution.
3. 5x + 5 = 5x + 4
Subtracting 5x from both sides:
5 = 4
This equation has no solution.
We can notice that the equations with no solution have the same coefficients and constant terms on both sides of the equation. This means that the equations are equivalent and contradictory, leading to no solution.
Having no solution means that there is no value of the variable that simultaneously satisfies all the equations in the system. It indicates that the equations are inconsistent and do not intersect at any point.
Checking the original equations with some points:
1. 2(1) + 3 = 5 ≠ 2(1) + 7
2. 4(2) + 22 = 30 ≠ 8(2) + 10
3. 5(-1) + 5 = 0 ≠ 5(-1) + 4
Therefore, we have confirmed that the system of equations has no solution.