Consider the following system of equations

x+y=2x+ky=2x+y=2x+ky=2

for what values of kk does the system have infinitely many solutions.  

Question 2Answer

a.

-1

b.

1

c.

0

d.

2

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.

Explore how to determine the values of k in a system of equations to achieve infinitely many solutions. This detailed explanation covers the analysis of coefficients and conditions required for infinite solutions.

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