Quiz Questions: No Solution vs. Infinitely Many Solutions in Algebra 1

📚 Quick Study Guide

• 🔢 No Solution: An equation has no solution when, after simplification, you arrive at a false statement. For example, $2 = 3$ is a false statement. This means there is no value for the variable that makes the equation true.
• ♾️ Infinitely Many Solutions: An equation has infinitely many solutions when, after simplification, you arrive at a true statement. For example, $0 = 0$ is a true statement. This means any value for the variable will make the equation true.
• ⚖️ Solving Equations: Use inverse operations to isolate the variable. Remember to perform the same operation on both sides of the equation to maintain balance.
• ➕ Combining Like Terms: Simplify each side of the equation by combining like terms before attempting to isolate the variable.
• ➗ Distributive Property: Use the distributive property to eliminate parentheses: $a(b + c) = ab + ac$.

Practice Quiz

1. Which of the following equations has no solution?
   1. A) $2x + 3 = 2x + 5$
   2. B) $3x + 4 = 4x + 4$
   3. C) $5x - 2 = 5x - 2$
   4. D) $x + 1 = x + 1$


2. Which of the following equations has infinitely many solutions?
   1. A) $4x - 4 = 4x + 4$
   2. B) $2(x + 1) = 2x + 2$
   3. C) $3x + 5 = 4x + 5$
   4. D) $x - 3 = x + 3$


3. Solve the equation: $3(x - 2) = 3x + 5$. What type of solution does it have?
   1. A) One solution
   2. B) No solution
   3. C) Infinitely many solutions
   4. D) Two solutions


4. Solve the equation: $2x + 4 = 2(x + 2)$. What type of solution does it have?
   1. A) One solution
   2. B) No solution
   3. C) Infinitely many solutions
   4. D) Two solutions


5. Which equation results in a false statement after simplification?
   1. A) $x + x = 2x$
   2. B) $3x - 1 = 3x + 1$
   3. C) $4 + x = x + 4$
   4. D) $0 = 0$


6. Which equation results in a true statement after simplification?
   1. A) $5x = 5x + 1$
   2. B) $2x + 3 = 2x - 3$
   3. C) $7x - 7 = 7(x - 1)$
   4. D) $x + 1 = x - 1$


7. What value of $k$ would make the equation $4x + 5 = 4x + k$ have no solution?
   1. A) $k = 5$
   2. B) $k = 0$
   3. C) $k = 6$
   4. D) $k = -5$