To determine the values of $k$ for which the system has infinitely many solutions, let's first analyze the given system of equations:

$x + y = 2 \quad \text{and} \quad 2x + ky = 2$

For the system to have infinitely many solutions, the two equations must be equivalent, meaning one equation must be a multiple of the other.

Step 1: Rewrite the equations

Let's express both equations in a similar format:

1. $x + y = 2$
2. $2x + ky = 2$

Step 2: Compare the coefficients

We need the ratio of the coefficients of $x$, $y$, and the constant term in both equations to be the same for the equations to be equivalent.

• The first equation is $x + y = 2$, which can be written as $1 \cdot x + 1 \cdot y = 2$.
• The second equation is $2x + ky = 2$, which is already in the correct form.

The ratio of the coefficients must be:

$\frac{1}{2} = \frac{1}{k} = \frac{2}{2}$

Step 3: Solve for $k$

For these ratios to be equal:

$\frac{1}{2} = \frac{1}{k}$

Cross-multiplying gives:

$k = 2$

Conclusion:

The system will have infinitely many solutions when $k = 2$. Thus, the correct answer is:

d. 2

Would you like more details or have any questions?

Here are five related questions:

1. How would you solve a system of equations if it had no solutions?
2. Can a system of equations be inconsistent? What does that mean?
3. How do you graphically interpret a system of equations with infinitely many solutions?
4. How does the value of $k$ change the nature of the system (unique solution, no solution, or infinitely many solutions)?
5. What happens if $k = 0$ in this system? Would it have a solution?

Tip: To determine the number of solutions in a system, always compare the ratios of the coefficients of the variables and the constant terms.