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

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(4 points)
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y = 5x + 7

3y - 15x = 18

y = 3x + 6

y = -13x - 4

x - 2y = 6

3x - 6y = 18

y = 23x - 1

y = 23x - 2

1 answer

To determine the number of solutions in the given system of equations, we need to analyze the relationships between the lines represented by each equation. The critical points to find out are:

  1. No Solutions: The lines are parallel (same slope, different intercept).
  2. One Solution: The lines intersect at a single point (different slopes).
  3. Infinitely Many Solutions: The lines are the same line (identical).

Let's analyze the provided equations:

  1. y = 5x + 7

    • Slope = 5

  2. 3y - 15x = 18

    • Rearranging gives: y = 5x + 6
    • Slope = 5 (parallel to the first equation)

  3. y = 3x + 6

    • Slope = 3 (different from the first two)

  4. y = -13x - 4

    • Slope = -13 (different from the first two)

  5. x - 2y = 6

    • Rearranging gives: 2y = x - 6 → y = (1/2)x - 3
    • Slope = 1/2 (different from the first two)

  6. 3x - 6y = 18

    • Rearranging gives: 6y = 3x - 18 → y = (1/2)x - 3
    • Slope = 1/2 (identical to fifth equation)

  7. y = 23x - 1

    • Slope = 23 (different from prior lines)

  8. y = 23x - 2

    • Slope = 23 (parallel to the previous equation)

Now, pairing them up based on the number of solutions:

  • No Solutions:

    • y = 5x + 7 (first equation) and 3y - 15x = 18 (which rearranges to y = 5x + 6):
    • Both of these have the same slope (5) but different y-intercepts (7 and 6).

  • One Solution:

    • Any pairing of different slopes leads to one solution. For example:
    • y = 3x + 6 and any other equation with a different slope, like y = -13x - 4.

  • Infinitely Many Solutions:

    • x - 2y = 6 and 3x - 6y = 18:
    • Both simplify to the same line (y = (1/2)x - 3), hence they have infinitely many solutions.

Here’s how they categorize:

  • No Solutions:

    • y = 5x + 7 and 3y - 15x = 18

  • Infinitely Many Solutions:

    • x - 2y = 6 and 3x - 6y = 18

  • One Solution:

    • Any combination of equations with different slopes like:
      • y = 3x + 6 and y = -13x - 4
      • y = 23x - 1 and y = 23x - 2

In summary, the equations can be paired as follows:

  • No Solutions: (y = 5x + 7, 3y - 15x = 18)
  • Infinitely Many Solutions: (x - 2y = 6, 3x - 6y = 18)
  • One Solution: (Any equations with different slopes.)