Question

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

Answer 1

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