Solve each system of equations on the left. Match the system to the correct number of solutions.(3 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. 1)2x-2y=-4 y=x+2

Question

2)y=-2x-7 y=-2x+3
3)y=3x-2 2x+3y=16

Answer 1

Let's solve the given systems of equations one by one:

1. System:
   \( 2x - 2y = -4 \)
   \( y = x + 2 \)

   First, we can rearrange the first equation: \[ 2x - 2y = -4 \implies 2y = 2x + 4 \implies y = x + 2 \] This shows that both equations are actually the same line, meaning there are infinitely many solutions (the lines coincide).

2. System:
   \( y = -2x - 7 \)
   \( y = -2x + 3 \)

   Since both equations have the same slope (-2), we can see that they are parallel but have different y-intercepts. Therefore, they do not intersect at any point, which means there are no solutions.

3. System:
   \( y = 3x - 2 \)
   \( 2x + 3y = 16 \)

   Let's substitute \( y \) from the first equation into the second: \[ 2x + 3(3x - 2) = 16 \implies 2x + 9x - 6 = 16 \implies 11x - 6 = 16 \implies 11x = 22 \implies x = 2 \] Then substitute \( x \) back into the first equation to find \( y \): \[ y = 3(2) - 2 = 6 - 2 = 4 \] The solution is \( (2, 4) \), which means there is exactly one solution.

Now summarizing the number of solutions for each system:

1. Infinitely many solutions
2. No solutions
3. One solution

Final Matches:

• 1. Infinitely many solutions
• 2. No solutions
• 3. One solution