Question

Match the system with the number of solutions.(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
y=5x−2 and y=5x+4
y=5x+1 and 2y=10x+2
y=x−3 and y=4x

Answer 1

To determine the number of solutions for each system of equations, we need to analyze the relationships between the lines represented by the equations.

1. **System 1:**
- \(y = 5x - 2\)
- \(y = 5x + 4\)

Both equations have the same slope (5), which means they are parallel lines. Since they have different y-intercepts, they will never intersect. Therefore, this system has **no solutions**.

2. **System 2:**
- \(y = 5x + 1\)
- \(2y = 10x + 2\) → This simplifies to \(y = 5x + 1\).

Both equations are actually the same line, meaning they overlap entirely. This system has **infinitely many solutions**.

3. **System 3:**
- \(y = x - 3\)
- \(y = 4x\)

These two equations have different slopes (1 for the first and 4 for the second), indicating that the lines will intersect at one point. Therefore, this system has **one solution**.

Here’s the summary of the matches:

- \(y = 5x - 2\) and \(y = 5x + 4\) → No solutions
- \(y = 5x + 1\) and \(2y = 10x + 2\) → Infinitely many solutions
- \(y = x - 3\) and \(y = 4x\) → One solution

So you would match them as follows (in the context of appropriate inputs):

1. **No solutions:** y=5x−2 and y=5x+4
2. **Infinitely many solutions:** y=5x+1 and 2y=10x+2
3. **One solution:** y=x−3 and y=4x