Determine the number of solutions for each equation:
a)x+3+7= 2x-10-x

The correct answer is: The given equations a)x + 3 + 7= 2x-10-x; b)3(x+4)= 3x+12 and c)5x+8= 2x-1 have No solution, Infinitely many solutions and 1 solution, respectively.

Step-by-step solution:-
a)x+3+7= 2x-10-x
Simplifying the given equation i.e. x+3+7= 2x - 10 - x, we get-
x + 3 + 7 = 2x - 10 - x
∴ x - 2x + x = - 10 - 3 - 7
∴ 0 ≠ - 20
∴ LHS ≠ RHS
Since in the above equation, LHS is not equal to RHS, the given equation has no solutions.
b)3(x+4)= 3x + 12
Simplifying the given equation i.e. 3(x+4)= 3x + 12, we get-
3 (x + 4) = 3x + 12
∴ 3x + 12 = 3x + 12
∴ LHS = RHS
Since, after simplifying the above equation, we get both sides of the equation equal to each other, the given equation has infinitely many solutions.
c)5x + 8 = 2x - 1
Simplifying the given equation i.e. 5x + 8 = 2x - 1, we get-
5x + 8 = 2x - 1
∴ 5x - 2x = - 1 - 8 .............................. (Taking variables and constants on either sides of the equation)
∴ 3x = -9
∴ x = - 3 ................................. (Dividing both sides by 3)
Since after simplifying the given equation, we get a single value of x i.e. -3 that will satisfy the given equation, we can conclude that the given equation has 1 solution.
Final Answer:-
∴ The given equations a)x + 3 + 7= 2x-10-x; b)3(x+4)= 3x+12 and c)5x+8= 2x-1 have No solution, Infinitely many solutions and 1 solution, respectively.