Final answer:
To find n and z for the system to have infinite solutions, one equation must be a multiple of the other. By analyzing and setting the coefficients of x and y equal after accounting for the multiples, we find that n = 2 and z = 6 are the needed values.
Step-by-step explanation:
To determine the values of n and z so that the system of equations has infinite solutions, we need to look for a relationship where one equation is a multiple of the other. The system of equations given is:
- 2(n + 1)x + 4y = 7
- 6x + (z - 2)y = 14
For the system to have infinite solutions, the ratios of the coefficients of x, y, and the constant terms must be equal in both equations. Let's analyze the equations:
- Equation 1, when simplified, is (2n + 2)x + 4y = 7.
- Equation 2 is already simplified: 6x + (z - 2)y = 14.
To have the same ratio, the coefficient of x in Equation 1 should be tripled to match the coefficient of x in Equation 2. Therefore, 2n + 2 must be equal to 6, which gives us n = 2. Similarly, the coefficient of y should also be the same after accounting for the multiples so, z - 2 must equal to 4, which gives us z = 6.