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).