The table shows two values of t and their corresponding values of T for a linear relationship.tT-7-251123The graph of the...

1. TRANSLATE the problem information

• Given information:
  • Table shows two points on a linear relationship: \(\mathrm{(-7, -25)}\) and \(\mathrm{(11, 23)}\)
  • The line passes through point \(\mathrm{(3/8, b)}\)
  • Need to find the value of b

• What this tells us: We need to find the equation of the line, then substitute \(\mathrm{t = 3/8}\) to find b

2. INFER the solution approach

• Since we have a linear relationship, we can write it as \(\mathrm{T = mt + c}\)
• To find this equation, we need the slope (m) and y-intercept (c)
• Strategy: Find slope using the two given points, then find c, then substitute

3. SIMPLIFY to find the slope

• Using slope formula with points \(\mathrm{(-7, -25)}\) and \(\mathrm{(11, 23)}\):
  \(\mathrm{m = \frac{23 - (-25)}{11 - (-7)}}\)
  \(\mathrm{m = \frac{48}{18}}\)
  \(\mathrm{m = \frac{8}{3}}\)

4. SIMPLIFY to find the y-intercept

• Using point \(\mathrm{(11, 23)}\) in \(\mathrm{T = mt + c}\):
  \(\mathrm{23 = \frac{8}{3}(11) + c}\)
  \(\mathrm{23 = \frac{88}{3} + c}\)
  \(\mathrm{c = 23 - \frac{88}{3}}\)

• Convert 23 to thirds:
  \(\mathrm{23 = \frac{69}{3}}\)
  \(\mathrm{c = \frac{69}{3} - \frac{88}{3}}\)
  \(\mathrm{c = -\frac{19}{3}}\)

5. SIMPLIFY to find b

• The equation is \(\mathrm{T = \frac{8}{3}t - \frac{19}{3}}\)
• When \(\mathrm{t = 3/8}\):
  \(\mathrm{b = \frac{8}{3} \times \frac{3}{8} - \frac{19}{3}}\)
  \(\mathrm{b = 1 - \frac{19}{3}}\)
  \(\mathrm{b = \frac{3}{3} - \frac{19}{3}}\)
  \(\mathrm{b = -\frac{16}{3}}\)

Answer: \(\mathrm{-\frac{16}{3}}\)

Alternative acceptable forms: \(\mathrm{-16/3}\), \(\mathrm{-5\frac{1}{3}}\), -5.333...

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when working with fractions, particularly when converting between whole numbers and fractions or combining fractions with different denominators.

Common mistakes include:

• Converting 23 incorrectly (writing \(\mathrm{23 = 23/3}\) instead of \(\mathrm{69/3}\))
• Sign errors when subtracting fractions
• Incorrectly multiplying \(\mathrm{\frac{8}{3} \times \frac{3}{8}}\)

This leads to getting a wrong final answer and confusion about whether they made a calculation error or approached the problem incorrectly.

Second Most Common Error:

Poor INFER reasoning: Students try shortcuts like using point-slope form directly with \(\mathrm{(3/8, b)}\) without recognizing that b is unknown, or they attempt to use the two given points to somehow directly find b without establishing the full linear equation first.

This causes them to get stuck early in the problem and resort to guessing rather than following the systematic approach.

The Bottom Line:

This problem requires careful fraction arithmetic throughout multiple steps. Students who are comfortable with fraction operations and follow the standard linear equation process (find slope, find intercept, substitute) will succeed, while those who struggle with fractions or try to skip steps often get lost.