y = 2/3x + 3 One of the two equations in a system of linear equations is given. The system...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{y = \frac{2}{3}x + 3}\)
  • System has infinitely many solutions
  • Second equation has form: \(\mathrm{y = mx + b}\)
• We need to find the value of b

2. INFER what "infinitely many solutions" means

• When a system of two linear equations has infinitely many solutions, it means the two equations represent the exact same line
• For two lines to be identical, they must have the same slope AND the same y-intercept
• This is the key insight that unlocks the entire problem

3. TRANSLATE the slope and y-intercept from the given equation

• From \(\mathrm{y = \frac{2}{3}x + 3}\), we can identify:
  • Slope \(\mathrm{(m) = \frac{2}{3}}\)
  • y-intercept \(\mathrm{(b) = 3}\)

4. INFER the requirements for the second equation

• Since both equations must represent the same line:
  • The second equation must have slope \(\mathrm{m = \frac{2}{3}}\)
  • The second equation must have y-intercept \(\mathrm{b = 3}\)
• Therefore, \(\mathrm{b = 3}\)

Answer: D. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't understand what "infinitely many solutions" means in the context of linear systems. They might think it means the equations are different but somehow work together, rather than recognizing they must be identical equations.

This conceptual gap leads to confusion about what values m and b should have, causing them to guess randomly among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly understand that the equations must be the same, but make errors reading the slope and y-intercept from \(\mathrm{y = \frac{2}{3}x + 3}\). They might confuse which number represents the slope versus the y-intercept.

This may lead them to select Choice A (-3) if they incorrectly think the y-intercept is negative, or get confused about the structure of slope-intercept form.

The Bottom Line:

This problem tests whether students understand the geometric meaning of "infinitely many solutions" - that two equations describe the same line. Once this connection is made, the solution becomes straightforward.