Interactive quiz: identify one, no, or infinite solutions from line graphs

📚 Quick Study Guide

• 📈 One Solution: The lines intersect at one point. This means the system has a unique solution corresponding to the coordinates of that intersection point.
• 🚫 No Solution: The lines are parallel and never intersect. This indicates that the system is inconsistent and has no solution. They have the same slope but different y-intercepts.
• ♾️ Infinite Solutions: The lines are identical, meaning they overlap completely. This shows that the system is dependent, and any point on the line is a solution. The equations are multiples of each other.
• 📐 Slope-Intercept Form: Remember that a linear equation can be written in slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• ✍️ Parallel Lines: Parallel lines have the same slope ($m$) but different y-intercepts ($b$).
• 겹쳐진 Overlapping Lines: Overlapping lines have the same slope ($m$) and the same y-intercept ($b$).

Practice Quiz

1. Which of the following scenarios indicates a system of linear equations with exactly one solution?

   1. The lines are parallel.
   2. The lines are perpendicular.
   3. The lines are overlapping.
   4. The lines never meet.



2. What is true about the slopes and y-intercepts of two lines that represent a system with infinite solutions?

   1. Different slopes, different y-intercepts.
   2. Same slope, different y-intercepts.
   3. Different slopes, same y-intercept.
   4. Same slope, same y-intercept.



3. If two lines on a graph are parallel, how many solutions does the corresponding system of equations have?

   1. One solution
   2. No solution
   3. Infinite solutions
   4. Two solutions



4. Which of the following equations, when graphed with $y = 2x + 3$, would result in a system with no solution?

   1. $y = 2x + 5$
   2. $y = -2x + 3$
   3. $y = 3x + 3$
   4. $2y = 4x + 6$



5. Which of the following graphs represents a system of equations with infinite solutions?

   1. Two distinct lines intersecting at one point.
   2. Two parallel lines.
   3. Two identical lines overlapping.
   4. Two perpendicular lines.



6. Consider the system: $y = 3x - 1$ and $y = 3x - 1$. How many solutions exist?

   1. One solution
   2. No solution
   3. Infinite solutions
   4. Two solutions



7. Two lines intersect at the point (2, 4). How many solutions does the system of equations represented by these lines have?

   1. Zero
   2. One
   3. Two
   4. Infinite