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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. 2x-2y=-4 y=x+2 y=-2x-7 y=-2x+3 y=3x-2 2x+3y=16

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To solve the given systems and match them to the number of solutions, we will analyze each system of equations one by one.

System 1:

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

Let's rewrite the first equation in slope-intercept form:

\[ 2x - 2y = -4 \ -2y = -2x - 4 \ y = x + 2 \]

Both equations represent the same line, therefore this system has infinitely many solutions (the lines are coincident).

System 2:

  1. \( 2x + 3y = 16 \)
  2. \( y = -2x + 3 \)

Let's substitute \( y \) from the second equation into the first equation:

\[ 2x + 3(-2x + 3) = 16 \ 2x - 6x + 9 = 16 \ -4x + 9 = 16 \ -4x = 16 - 9 \ -4x = 7 \ x = -\frac{7}{4} \]

Now find \( y \):

\[ y = -2\left(-\frac{7}{4}\right) + 3 \ y = \frac{14}{4} + 3 = \frac{14}{4} + \frac{12}{4} \ y = \frac{26}{4} = \frac{13}{2} \]

So the solution is \( (-\frac{7}{4}, \frac{13}{2}) \), indicating this system has one solution.

Summary of Solutions:

  • System 1 (2x - 2y = -4 and y = x + 2) has infinitely many solutions.
  • System 2 (2x + 3y = 16 and y = -2x + 3) has one solution.

Matching to responses:

  • For infinitely many solutions: Match with \( y = x + 2 \).
  • For one solution: Match with \( y = -2x + 3 \) or \( 2x + 3y = 16 \).

You can now insert these matches into the correct inputs, depending on how the tool you are using is set up.