Viktoriya solved an equation as shown below.

Question

5 (x minus 2) minus 3 x = 2 (x minus 3) minus 16. 5 x minus 10 minus 3 x = 2 x minus 6 minus 16. 2 x minus 10 = 2 x minus 22. negative 10 = negative 22.

She says that the equation has infinitely many solutions. Which describes Viktoriya’s error?
Viktoriya used the distributive property incorrectly when she found the solution.
Viktoriya should have said that the equation has no solution.
Viktoriya should have continued to solve the equation to find that x = 0.
Viktoriya should have said that the equation has one unique solution.

Answer 1

Viktoriya's error lies in her conclusion after simplifying the equation. Let's analyze the steps she took:

1. Original Equation: \( 5(x - 2) - 3x = 2(x - 3) - 16 \)

2. Distributing: \( 5x - 10 - 3x = 2x - 6 - 16 \)

3. Combining Like Terms:

   • Left side: \( 5x - 3x = 2x \) → becomes \( 2x - 10 \)
   • Right side: \( -6 - 16 = -22 \) → becomes \( 2x - 22 \)

   Now it simplifies to: \( 2x - 10 = 2x - 22 \)

4. Subtracting \( 2x \) from both sides: This leads to: \( -10 = -22 \)

This final statement is a contradiction, indicating that there is no solution to the equation.

So, Viktoriya's error is in incorrectly concluding that the equation has infinitely many solutions. Instead, based on her simplification which led to a contradiction, she should have said that the equation has no solution.

Therefore, the correct choice describing Viktoriya's error is: Viktoriya should have said that the equation has no solution.