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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

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options: one solution, no solution, infinite solutions

1 answer

To determine the number of solutions for each system of equations, we need to analyze each pair of equations. Here’s how they match up:

  1. y = 5x + 7 and 3y - 15x = 18:

    • Rewrite the second equation in slope-intercept form: \(3y = 15x + 18 \Rightarrow y = 5x + 6\).
    • These lines are parallel and not the same (different y-intercepts), which means there is no solution.

  2. y = 3x + 6 and y = -13x - 4:

    • The slopes (3 and -13) are different, indicating they intersect at one point. Thus, there is one solution.

  3. x - 2y = 6 and 3x - 6y = 18:

    • Rewrite the first equation: \(2y = x - 6 \Rightarrow y = \frac{1}{2}x - 3\).
    • The second equation can be simplified to: \(x - 2y = 6\).
    • Since this is just a rearrangement of the first, these lines are the same. Therefore, there are infinite solutions.

  4. y = 23x - 1 and y = 23x - 2:

    • The slopes are the same (23) and the y-intercepts are different, indicating that the lines are parallel. Thus, there is no solution.

  5. y = 5x + 7 and 3x - 6y = 18 (reduction to y = ...):

    • As noted earlier, this setup also yields no solution, since they are essentially parallel.

In summary, the equations correspond to:

  • One Solution: y = 3x + 6 and y = -13x - 4
  • No Solution: y = 5x + 7 and 3y - 15x = 18 / y = 23x - 1 and y = 23x - 2
  • Infinite Solutions: x - 2y = 6 and 3x - 6y = 18

The corresponding matches are:

  • One solution: y = 3x + 6, y = -13x - 4
  • No solution: y = 5x + 7, 3y - 15x = 18
  • Infinite solutions: x - 2y = 6, 3x - 6y = 18

You would fill in:

  • One solution with y = 3x + 6 and y = -13x - 4.
  • No solution with y = 5x + 7 and 3y - 15x = 18.
  • Infinite solutions with x - 2y = 6 and 3x - 6y = 18.