Does an equation always have a solution

No, there is not always a solution to every equation. Infinitely many solutions is when both sides have the same equation. Ex: 4x - 1 = 4x - 1  No matter what you plug in.. it will be equal on both sides. 

No solution is something like this: 4x=4x+1If you subtract the 4x, you get 0 = -1
That is obviously not equal to -1, so there is no solution.
No matter what number you plug in for x, it will NOT be equal on both sides. 

To put it shortly:
Infinitely many solutions- EVERYTHING works
No solution- NOTHING works

m∠cdf = 43° , m∠gdf = 114°, m∠gde = 23°, m∠edh = 23° , m∠cdh = 157°

step-by-step explanation:

since, it is given that de bisects the angle gdh ⇒ de divides the angle gdh into two equal angles.

⇒ m∠gde = m∠edh

⇒ 8·x   - 1 = 6·x + 5

⇒ x = 3

so, m∠gde = 8·x - 1 = 23°

and m∠edh = 23°

now, it is given ∠cde is an straight angle so, m∠cde = 180°

⇒ m∠cdf + m∠fdg + m∠gde = 180°

⇒ 43° + m∠fdg + 23° = 180°

⇒ m∠fdg = 114°

again, m∠cde = m∠cdh + m∠edh = 180°

m∠cdh + 23° = 180°

⇒ m∠cdh = 157°