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Match each system on the left with all words that describe the system on the right. Choices on the right can be used more than once.

y=2x+3
x+y=-3
3y=9x-6
2y-6x=4
y=-1/2x+2
x+2y=4
inconsistent
consistent
independent
dependent

1 answer

To categorize the given systems of equations as inconsistent, consistent, independent, or dependent, we first need to understand what each term means:

  • Inconsistent: A system that has no solutions (the lines representing the equations do not intersect).
  • Consistent: A system that has at least one solution (the lines intersect at least once).
  • Independent: A consistent system where the lines intersect at exactly one point.
  • Dependent: A consistent system where the lines overlap, leading to infinitely many solutions.

We'll analyze each equation and determine how they relate to one another.

  1. y = 2x + 3

    • This is a line with a slope of 2. It can be represented as an equation in standard form for comparison with others.

  2. x + y = -3

    • This can be rewritten as \(y = -x - 3\). The slope is -1.

  3. 3y = 9x - 6

    • Simplifying yields \(y = 3x - 2\) (slope of 3).

  4. 2y - 6x = 4

    • Rearranging gives \(y = 3x + 2\) (slope of 3).

  5. y = -1/2x + 2

    • This slope is -1/2.

  6. x + 2y = 4

    • Rearranging gives \(y = -0.5x + 2\) (slope of -0.5).

Now, let's compare the slopes of the equations:

  • Equations 3 and 4 have the same slope of 3 and can be expressed in a dependent form, as they yield the same line. Therefore, they are dependent.
  • The equations y = 2x + 3, y = -x - 3 (2), y = -1/2x + 2 (5), and y = -0.5x + 2 (6) all have different slopes, indicating they are independent.

Now, let's categorize:

  • y = 2x + 3: Independent, consistent (intersects with the next equations).
  • x + y = -3: Independent, consistent (crosses other lines).
  • 3y = 9x - 6: Dependent (because it is the same line as 2y - 6x = 4).
  • 2y - 6x = 4: Dependent (same line as above).
  • y = -1/2x + 2: Independent, consistent.
  • x + 2y = 4: Independent, consistent.

Finally, we result in:

  1. y = 2x + 3: consistent, independent
  2. x + y = -3: consistent, independent
  3. 3y = 9x - 6: consistent, dependent
  4. 2y - 6x = 4: consistent, dependent
  5. y = -1/2x + 2: consistent, independent
  6. x + 2y = 4: consistent, independent

So the matches are as follows:

  • y = 2x + 3: consistent, independent
  • x + y = -3: consistent, independent
  • 3y = 9x - 6: consistent, dependent
  • 2y - 6x = 4: consistent, dependent
  • y = -1/2x + 2: consistent, independent
  • x + 2y = 4: consistent, independent