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Match the system of equations to the number of solutions it has.

(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.

y=5x+7
3y-15x=21

y=-13x-4
y=3x+6

y=5x-1
y=5x+2

responces:
no solution
one solution
infinite solutions

1 answer

Let's analyze each system of equations to determine the number of solutions.

  1. First system:

    • \(y = 5x + 7\)
    • \(3y - 15x = 21\)

    Rewrite the second equation: \[ 3y = 15x + 21 \quad \Rightarrow \quad y = 5x + 7 \] Since both equations are equivalent, they have infinite solutions.

  2. Second system:

    • \(y = -13x - 4\)
    • \(y = 3x + 6\)

    Since the slopes are different (-13 and 3), the graphs of these lines will intersect at exactly one point. Therefore, this system has one solution.

  3. Third system:

    • \(y = 5x - 1\)
    • \(y = 5x + 2\)

    Both equations have the same slope (5), but different y-intercepts (-1 and 2). Therefore, these lines are parallel and will never intersect. This means there is no solution.

Now we can insert the responses:

  • First system: infinite solutions
  • Second system: one solution
  • Third system: no solution