Final answer:
The system of equations can be solved using the elimination method, yielding the solution y = 4 and z = -3 after eliminating y and substituting z back into one of the original equations.
Step-by-step explanation:
To solve the system of equations consisting of 4y+3z=7 (equation a) and 5z-4y=-31 (equation b), we can use a method such as substitution or elimination. Since the coefficients of y in both equations are opposites, elimination is a convenient method to use here. Adding the two equations directly simplifies the system because the y terms will cancel out.
Performing the addition, we get:
4y + 3z + (-4y + 5z) = 7 - 31
This simplifies to:
8z = -24
Dividing both sides by 8 gives us z = -3.
Once we have the value of z, we can substitute it back into one of the original equations to find y. Using equation a, we get:
4y + 3(-3) = 7
4y - 9 = 7
Adding 9 to both sides gives:
4y = 16
Dividing by 4 gives y = 4.
So the solution to the system is y = 4 and z = -3.