Examples of systems of equations with infinite solutions

📚 Quick Study Guide

• 🔢 A system of equations has infinite solutions when the equations represent the same line.
• 🧑‍🏫 This means one equation is a multiple of the other after simplification. For example, $2x + 2y = 4$ and $x + y = 2$ represent the same line.
• ➗ To check, simplify both equations. If they are identical, there are infinite solutions.
• 📈 Graphically, both equations plot as the exact same line.
• 📝 A dependent system will always have infinite solutions.
• 🧮 In matrix form, infinite solutions occur when you have a row of zeros after row reduction and the corresponding entry in the augmented column is also zero.

Practice Quiz

1. Which of the following systems of equations has infinite solutions?

   1. $x + y = 5$, $x - y = 1$
   2. $2x + 2y = 6$, $x + y = 3$
   3. $x + y = 4$, $x + y = 8$
   4. $x = 2$, $y = 3$

2. Which system represents the same line?

   1. $x + y = 1$, $x - y = 1$
   2. $3x + 3y = 9$, $x + y = 4$
   3. $4x + 4y = 8$, $x + y = 2$
   4. $x = 5$, $y = 6$

3. For what value of $k$ will the following system have infinite solutions? $x + y = 3$, $2x + 2y = k$

   1. $k = 2$
   2. $k = 3$
   3. $k = 6$
   4. $k = 9$

4. Which system has dependent equations?

   1. $x + y = 7$, $x - y = 1$
   2. $x + y = 10$, $2x + y = 5$
   3. $3x + 3y = 12$, $x + y = 4$
   4. $x = 1$, $y = 2$

5. Which system of equations represents the same line graphically?

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

6. Which of the following is an example of a system with infinite solutions?

   1. $x + y = 0, x - y = 0$
   2. $x + y = 1, x = 2$
   3. $x - y = 1, 2x - 2y = 2$
   4. $x = 3, y = 4$

7. Determine the value of 'a' such that the system has infinite solutions: $x + y = 5$, $ax + ay = 15$

   1. $a = 1$
   2. $a = 2$
   3. $a = 3$
   4. $a = 4$