Consider this system of linear equations:

y = –3x + 5

y = mx + b

Which values of m and b will create a system of linear equations with no solution? 

m = –3 and b = –3
m = 5 and b = –3
m = 3 and b = 5
m = -3 and b = 5

To determine which values of \( m \) and \( b \) will result in a system of linear equations with no solution, we need to analyze the given equations:

1. **First equation:** \( y = -3x + 5 \)
2. **Second equation:** \( y = mx + b \)

### Condition for No Solution:
A system of linear equations has no solution when the lines are **parallel**. Parallel lines have the **same slope** but different y-intercepts.

#### Step 1: Slope and intercept of the first equation
- The slope (\( m_1 \)) of the first equation is \( -3 \).
- The y-intercept (\( b_1 \)) of the first equation is \( 5 \).

#### Step 2: Analyze the options for \( m \) and \( b \):
- If the second equation has a slope (\( m_2 \)) equal to \( -3 \), the lines will be parallel if their y-intercepts (\( b \)) are different.

#### Step 3: Evaluate each option:
1. \( m = -3 \) and \( b = -3 \):  
   - Slope: \( m_2 = -3 \), which matches \( m_1 \).
   - Intercept: \( b_2 = -3 \), which is different from \( b_1 = 5 \).  
   - **Result:** No solution (lines are parallel).

2. \( m = 5 \) and \( b = -3 \):  
   - Slope: \( m_2 = 5 \), which does not match \( m_1 = -3 \).  
   - **Result:** Not parallel (intersecting lines).

3. \( m = 3 \) and \( b = 5 \):  
   - Slope: \( m_2 = 3 \), which does not match \( m_1 = -3 \).  
   - **Result:** Not parallel (intersecting lines).

4. \( m = -3 \) and \( b = 5 \):  
   - Slope: \( m_2 = -3 \), which matches \( m_1 = -3 \).
   - Intercept: \( b_2 = 5 \), which is the same as \( b_1 = 5 \).  
   - **Result:** Infinite solutions (lines are the same).

### Final Answer:
The values of \( m \) and \( b \) that create a system of linear equations with no solution are:
\[
\boxed{m = -3 \text{ and } b = -3}
\]

Let me know if you'd like further clarification or calculations! 😊

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### 5 Related Questions:
1. What is the general condition for two lines to have infinite solutions?
2. How can you determine the point of intersection for two non-parallel lines?
3. What happens when two lines have different slopes?
4. How do you graph a system of linear equations with no solution?
5. Can two lines with the same slope and y-intercept ever have no solution?

### Tip:
To identify parallel lines quickly, focus on their slopes. Equal slopes mean they are parallel (or identical if intercepts match).

Consider this system of linear equations:

y = –3x + 5

y = mx + b

Which values of m and b will create a system of linear equations with no solution?

m = –3 and b = –3
m = 5 and b = –3
m = 3 and b = 5
m = -3 and b = 5

Explore how to determine the conditions under which a system of linear equations has no solution. This problem involves analyzing the slope-intercept form of equations and understanding parallel lines. Suitable for Grades 8-10 students studying algebra.

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