Question

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

(3 points)
Put responses in the correct input to answer the question.
y=5x+7

3y-15x=21

y=-13x-4

y=3x+6

y=5x-1

y=5x+2

Answer 1

To determine the number of solutions for the given system of equations, we should analyze the slopes and intercepts of the lines represented by each equation.

1. **y = 5x + 7**
- Slope = 5

2. **3y - 15x = 21**
- Rewriting gives: y = 5x + 7
- Slope = 5 (same as first equation)

3. **y = -13x - 4**
- Slope = -13 (different slope)

4. **y = 3x + 6**
- Slope = 3 (different slope)

5. **y = 5x - 1**
- Slope = 5 (same slope as 1st and 2nd equations)

6. **y = 5x + 2**
- Slope = 5 (same slope as 1st, 2nd, and 5th equations)

Now we categorize based on the slopes:
- Equations 1, 2, 5, and 6 have the same slope (5), meaning they are parallel and hence do not intersect. They do not share any solutions.
- Equations 3 and 4 have different slopes (-13 and 3 respectively). They will intersect with each other and with the equations with slope 5 at two distinct points (since they are not parallel to each other).

### Conclusion
Therefore, we can conclude:
- The equations with the same slope (y = 5x + 7; 3y - 15x = 21; y = 5x - 1; y = 5x + 2) have **zero solutions** among themselves.
- Thus, you have two distinct lines (which will intersect) represented by equations with different slopes.

### Final Matching
- Equations with zero solutions: **y = 5x + 7; 3y - 15x = 21; y = 5x - 1; y = 5x + 2**
- Equations that intersect (at least one solution): **y = -13x - 4; y = 3x + 6**

This gives you an overall understanding of the number of solutions for the various equations presented.