Ask a New Question
Menu

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

Put responses in the correct input to answer the question.

1. y = 5x + 7

3y - 15x = 21

2. y = -13x - 4

y = 3x + 6

3. y = 5x - 1

y = 5x + 2

1 answer

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

  1. First system:

    • Equation 1: \( y = 5x + 7 \)
    • Equation 2: \( 3y - 15x = 21 \)
      • Simplifying Equation 2: \[ 3y = 15x + 21 \] \[ y = 5x + 7 \]
      • This means both equations represent the same line, so there are infinitely many solutions.

  2. Second system:

    • Equation 1: \( y = -13x - 4 \)
    • Equation 2: \( y = 3x + 6 \)
      • These are two lines with different slopes (the first has a slope of -13 and the second has a slope of 3), so they will intersect at exactly one point, resulting in one solution.

  3. Third system:

    • Equation 1: \( y = 5x - 1 \)
    • Equation 2: \( y = 5x + 2 \)
      • Both lines have the same slope (5), but different y-intercepts (-1 and +2), indicating they are parallel and will never intersect. Thus, there are no solutions.

Now, summarizing the findings:

  • The first system has infinitely many solutions.
  • The second system has one solution.
  • The third system has no solutions.

Here’s the final arrangement of answers based on the systems:

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