Final answer:
The given equation is simplified to find the value of z, resulting in exactly one solution for the linear equation.
Step-by-step explanation:
The equation given is 3z + 9 + 14z = 4z + 5. To find out how many solutions the equation has, we first need to collect like terms on each side of the equation.
Combining the terms with z on the left-hand side, we get 17z + 9 = 4z + 5.
Next, we'll subtract 4z from both sides to get all the z terms on one side, resulting in 13z + 9 = 5.
Finally, subtract 9 from both sides to solve for z, leaving us with 13z = -4.
Dividing both sides by 13 gives us z = -4/13 or approximately z = -0.31.
Since this is a linear equation in one variable and we found a value for z, the equation has exactly one solution.