Question

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

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(4 points)
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y = 5x + 7

3y - 15x = 18

y = 3x + 6

y = -13x - 4

x - 2y = 6

3x - 6y = 18

y = 23x - 1

y = 23x - 2

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options: one solution, no solution, infinite solutions

Answer 1

To determine the number of solutions for each system of equations, we need to analyze each pair of equations. Here’s how they match up:

1. **y = 5x + 7** and **3y - 15x = 18**:
- Rewrite the second equation in slope-intercept form: \(3y = 15x + 18 \Rightarrow y = 5x + 6\).
- These lines are parallel and not the same (different y-intercepts), which means there is **no solution**.

2. **y = 3x + 6** and **y = -13x - 4**:
- The slopes (3 and -13) are different, indicating they intersect at one point. Thus, there is **one solution**.

3. **x - 2y = 6** and **3x - 6y = 18**:
- Rewrite the first equation: \(2y = x - 6 \Rightarrow y = \frac{1}{2}x - 3\).
- The second equation can be simplified to: \(x - 2y = 6\).
- Since this is just a rearrangement of the first, these lines are the same. Therefore, there are **infinite solutions**.

4. **y = 23x - 1** and **y = 23x - 2**:
- The slopes are the same (23) and the y-intercepts are different, indicating that the lines are parallel. Thus, there is **no solution**.

5. **y = 5x + 7** and **3x - 6y = 18** (reduction to `y = ...`):
- As noted earlier, this setup also yields **no solution**, since they are essentially parallel.

In summary, the equations correspond to:

- **One Solution:** **y = 3x + 6** and **y = -13x - 4**
- **No Solution:** **y = 5x + 7** and **3y - 15x = 18** / **y = 23x - 1** and **y = 23x - 2**
- **Infinite Solutions:** **x - 2y = 6** and **3x - 6y = 18**

The corresponding matches are:
- One solution: y = 3x + 6, y = -13x - 4
- No solution: y = 5x + 7, 3y - 15x = 18
- Infinite solutions: x - 2y = 6, 3x - 6y = 18

You would fill in:
- **One solution** with **y = 3x + 6** and **y = -13x - 4**.
- **No solution** with **y = 5x + 7** and **3y - 15x = 18**.
- **Infinite solutions** with **x - 2y = 6** and **3x - 6y = 18**.