Question

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

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(4 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 + 7

3y - 15x = 18

y = 3x + 6

y = -13x - 4

x - 2y = 6

3x - 6y = 18

y = 23x - 1

y = 23x - 2

Options: One Solution, No Solution, Infinite Solutions

Answer 1

To match each system of equations with the number of solutions it has, we need to analyze the equations provided. We'll determine if they have one solution, no solution, or infinite solutions based on their slopes and intercepts.

1. **y = 5x + 7**
- Slope: 5, Intercept: 7
- **Unique line** - could intersect with other lines.

2. **3y - 15x = 18** (Rearranged: y = 5x + 6)
- Slope: 5, Intercept: 6
- **Same slope as y = 5x + 7, different intercept**
- **No solution** (parallel lines).

3. **y = 3x + 6**
- Slope: 3, Intercept: 6
- **Unique line** - intersects with others unless lines are parallel.

4. **y = -13x - 4**
- Slope: -13, Intercept: -4
- **Different slope** from others; likely intersects.

5. **x - 2y = 6** (Rearranged: y = 0.5x - 3)
- Slope: 0.5, Intercept: -3
- **Unique line** - intersects with others unless parallel.

6. **3x - 6y = 18** (Rearranged: y = 0.5x - 3)
- Same as above - has the same slope and intercept as the previous equation.
- **Infinite solutions** (same line).

7. **y = 23x - 1**
- Slope: 23, Intercept: -1
- **Unique line** - intersects unless parallel.

8. **y = 23x - 2**
- Slope: 23, Intercept: -2
- **Same slope as above, different intercept**
- **No solution** (parallel).

Now we match each equation(s) with the number of solutions:

- **One Solution**:
- y = 5x + 7
- y = 3x + 6
- y = -13x - 4
- x - 2y = 6
- y = 23x - 1

- **No Solution**:
- 3y - 15x = 18 (same as y = 5x + 6)
- y = 23x - 2 (parallel to y = 23x - 1)

- **Infinite Solutions**:
- 3x - 6y = 18 (same line as x - 2y = 6)

So you can fill your answer with these matches:

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