Question

Match the system of equations to the number of solutions it has.
y=5x+7 3y-15x=21
y=-13x-4 y=3x+6
y=5x-1 y=5x+2

no solution
one solution
infinite solutions

Answer 1

To determine the number of solutions for each system of equations, we can analyze them:

1. **System:**
- \( y = 5x + 7 \)
- \( 3y - 15x = 21 \)

Let's rewrite the second equation:
\[
3y - 15x = 21 \implies y = 5x + 7
\]
This is identical to the first equation, so they are the same line.
- **Number of solutions:** Infinite solutions.

2. **System:**
- \( y = -13x - 4 \)
- \( y = 3x + 6 \)

These are two different linear equations with different slopes:
- The slope of \( y = -13x - 4 \) is -13.
- The slope of \( y = 3x + 6 \) is 3.

Since the slopes are different, the two lines will intersect at one point.
- **Number of solutions:** One solution.

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

Both equations have the same slope of 5:
- \( y = 5x - 1 \) (slope = 5)
- \( y = 5x + 2 \) (slope = 5)

Since the y-intercepts are different (-1 and +2), the lines are parallel and there is no intersection.
- **Number of solutions:** No solution.

### Final Summary:
- **First system:** Infinite solutions
- **Second system:** One solution
- **Third system:** No solution