Question

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

Put responses in the correct input to answer the question.

1. y = 5x + 7

3y - 15x = 21

2. y = -13x - 4

y = 3x + 6

3. y = 5x - 1

y = 5x + 2

Answer 1

Let's analyze each system of equations to determine the number of solutions:

1. **First system:**
- Equation 1: \( y = 5x + 7 \)
- Equation 2: \( 3y - 15x = 21 \)
- Simplifying Equation 2:
\[ 3y = 15x + 21 \]
\[ y = 5x + 7 \]
- This means both equations represent the same line, so there are infinitely many solutions.

2. **Second system:**
- Equation 1: \( y = -13x - 4 \)
- Equation 2: \( y = 3x + 6 \)
- These are two lines with different slopes (the first has a slope of -13 and the second has a slope of 3), so they will intersect at exactly one point, resulting in one solution.

3. **Third system:**
- Equation 1: \( y = 5x - 1 \)
- Equation 2: \( y = 5x + 2 \)
- Both lines have the same slope (5), but different y-intercepts (-1 and +2), indicating they are parallel and will never intersect. Thus, there are no solutions.

Now, summarizing the findings:
- The first system has **infinitely many solutions.**
- The second system has **one solution.**
- The third system has **no solutions.**

Here’s the final arrangement of answers based on the systems:

1. Infinitely many solutions
2. One solution
3. No solutions