Question

Match the system of equations to the number of solutions it has.(4 points)
Put responses in the correct input to answer the question.
1. y=5x+7 and 3y−15x=18:
2. x−2y=6 and 3x−6y=18:
3. y=23x−1 and y=23x−2:
4. 3x+6=y and y=−13x−4:

Answer 1

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

1. **y = 5x + 7** and **3y - 15x = 18:**
- The second equation can be simplified: \( 3y = 15x + 18 \) or \( y = 5x + 6 \).
- Both equations have the same slope (5) but different y-intercepts (7 vs 6), which indicates that they are parallel lines.
- **Number of solutions: 0.**

2. **x - 2y = 6** and **3x - 6y = 18:**
- The second equation can be simplified: \( 3x - 6y = 18 \) becomes \( x - 2y = 6 \) after dividing by 3.
- Since both equations are the same, they represent the same line.
- **Number of solutions: Infinite.**

3. **y = 23x - 1** and **y = 23x - 2:**
- Both equations have the same slope (23) and different y-intercepts (-1 vs -2). They are parallel lines.
- **Number of solutions: 0.**

4. **3x + 6 = y** and **y = -13x - 4:**
- The first equation can be rewritten as \( y = 3x + 6 \).
- The two equations have different slopes (3 vs -13), which means they will intersect at one point.
- **Number of solutions: 1.**

Putting them all together, we have:
- 1: 0 solutions
- 2: Infinite solutions
- 3: 0 solutions
- 4: 1 solution

Final response:
1. 0
2. Infinite
3. 0
4. 1