Consider the system of equations below.\(\mathrm{y = 3(2x - 5)}\)6y = 36x - 90How many solutions \(\mathrm{(x, y)}\) does this...

1. TRANSLATE the problem setup

• Given system:
  • \(\mathrm{y = 3(2x - 5)}\)
  • \(\mathrm{6y = 36x - 90}\)
• Need to find: How many solution pairs (x, y) exist

2. INFER the solution strategy

• Use substitution method: substitute the y-expression from equation 1 into equation 2
• This will reveal the relationship between the two equations

3. SIMPLIFY through substitution

• Substitute \(\mathrm{y = 3(2x - 5)}\) into the second equation:

\(\mathrm{6[3(2x - 5)] = 36x - 90}\)

• SIMPLIFY the left side using distributive property:

\(\mathrm{6(6x - 15) = 36x - 90}\)

\(\mathrm{36x - 90 = 36x - 90}\)

4. INFER the meaning of the result

• The equation \(\mathrm{36x - 90 = 36x - 90}\) is an identity (always true)
• This means the original equations are equivalent - they represent the same line
• When two equations represent the same line, every point on that line satisfies both equations

Answer: D) Infinitely many

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding what an identity means

Students correctly perform the substitution and arrive at \(\mathrm{36x - 90 = 36x - 90}\), but then think this means "no solution" because "nothing cancels out" or "we can't solve for x." They don't recognize that when both sides of an equation are identical, it means the equation is true for ALL values of the variable.

This may lead them to select Choice A (Zero).

Second Most Common Error:

Poor SIMPLIFY execution: Making distributive property errors

Students might incorrectly distribute when computing \(\mathrm{6[3(2x - 5)]}\), perhaps getting \(\mathrm{18x - 15}\) instead of \(\mathrm{36x - 90}\). This leads to a different equation that might seem to have exactly one solution or no solution.

This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

The key insight is recognizing that identical equations (or equations that simplify to an identity) represent the same line, and therefore share infinitely many points in common. Students often expect to "solve for" specific x and y values, but systems can have zero, one, or infinitely many solutions depending on how the lines relate to each other.