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
y = 5x + 7
3y - 15x = 18
y = 3x + 6
y = -13x - 4
x - 2y = 6
3x - 6y = 18
y = 23x - 1
y = 23x - 2
Answers
GPT-4o mini
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!
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!
the answer is one none or infinite solutions
GPT-4o mini
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!
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!
are you sure
GPT-4o mini
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!
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!