Question

Match the system of equations to the number of solutions it has
y = 5x + 7

3y - 15x = 18

y = 3x + 6

y = -13x - 4

x - 2y = 6

3x - 6y = 18

y = 23x - 1

y = 23x - 2

Answer 1

the answer is one none or infinite solutions

Answer 2

are you sure

Answer 3

To determine the number of solutions for the given system of equations, we need to analyze each pair of equations. Here’s the breakdown:

1. **Equation Pair:**
- \( y = 5x + 7 \)
- \( 3y - 15x = 18 \)

The second equation can be rewritten as:
\[
y = 5x + 6
\]
Since the slopes are the same but the y-intercepts are different, these lines are parallel. **Number of solutions: 0**.

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

These two lines have different slopes, so they will intersect at one point. **Number of solutions: 1**.

3. **Equation Pair:**
- \( x - 2y = 6 \)
- \( 3x - 6y = 18 \)

The second equation can be rewritten as:
\[
x - 2y = 6
\]
Since both equations are identical, they represent the same line. **Number of solutions: Infinite** (all points on the line).

4. **Equation Pair:**
- \( y = 23x - 1 \)
- \( y = 23x - 2 \)

The slopes are the same but the y-intercepts are different, meaning these lines are parallel. **Number of solutions: 0**.

Now, summarizing the solutions for the pairs:
- \( y = 5x + 7 \) and \( 3y - 15x = 18 \): 0 solutions (parallel)
- \( y = 3x + 6 \) and \( y = -13x - 4 \): 1 solution (intersect)
- \( x - 2y = 6 \) and \( 3x - 6y = 18 \): Infinite solutions (same line)
- \( y = 23x - 1 \) and \( y = 23x - 2 \): 0 solutions (parallel)

Please clarify if you meant to ask something else or if you'd like specific combinations rated for number of solutions!

Answer 4

Thank you for the clarification! Here are the conclusions for the given pairs of equations, summarized in terms of the number of solutions: one, none, or infinite solutions.

1. **Equations:**
- \(y = 5x + 7\)
- \(3y - 15x = 18\) (which simplifies to \(y = 5x + 6\))

**Number of Solutions:** **None** (parallel lines)

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

**Number of Solutions:** **One** (intersecting lines)

3. **Equations:**
- \(x - 2y = 6\)
- \(3x - 6y = 18\) (which simplifies to \(x - 2y = 6\))

**Number of Solutions:** **Infinite** (same line)

4. **Equations:**
- \(y = 23x - 1\)
- \(y = 23x - 2\)

**Number of Solutions:** **None** (parallel lines)

### Summary:
1. **None**
2. **One**
3. **Infinite**
4. **None**

If you have any additional questions or need further assistance, feel free to ask!

Answer 5

Let’s carefully analyze each pair of equations again step by step to ensure the correctness of the classifications for the number of solutions (one, none, or infinite).

1. **Equations:**
- \(y = 5x + 7\)
- \(3y - 15x = 18\)

**Step 1:** Rewrite the second equation in slope-intercept form:
\[
3y = 15x + 18 \implies y = 5x + 6
\]
**Conclusion:** The first equation is \(y = 5x + 7\) and the second is \(y = 5x + 6\). Both have the same slope but different y-intercepts. **Therefore, they are parallel and have** **none** **(0) solutions**.

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

**Conclusion:** The two lines have different slopes (3 vs. -13), meaning they intersect. Hence, there is **one solution**.

3. **Equations:**
- \(x - 2y = 6\)
- \(3x - 6y = 18\)

**Step 2:** Rewrite the first equation to check for equivalence:
\[
x - 2y = 6 \implies y = \frac{1}{2}x - 3
\]
**Step 3:** Rewrite the second equation:
\[
3x - 6y = 18 \implies -6y = -3x + 18 \implies y = \frac{1}{2}x - 3
\]
Both equations are identical, meaning every point on the line satisfies both equations. Therefore, there are **infinite solutions**.

4. **Equations:**
- \(y = 23x - 1\)
- \(y = 23x - 2\)

**Conclusion:** These lines have the same slope but different y-intercepts. Therefore, they are parallel and have **none (0) solutions**.

### Final Summary:
1. **None** (parallel lines)
2. **One** (intersecting lines)
3. **Infinite** (same line)
4. **None** (parallel lines)

So, the classifications are confirmed as follows:
- Pair 1: None
- Pair 2: One
- Pair 3: Infinite
- Pair 4: None

If you have any further questions or need clarifications, please feel free to ask!