For the linear function h, the graph of \(\mathrm{y = h(x)}\) in the xy-plane passes through the points \(\mathrm{(7, 21)}\)...

1. INFER the solution strategy

• Given: Linear function h passes through \((7, 21)\) and \((9, 25)\)
• Need: Equation \(\mathrm{h(x) = mx + b}\)
• Strategy: Find slope m first, then find y-intercept b

2. SIMPLIFY to find the slope

• Apply slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Substitute points: \(\mathrm{m = \frac{25 - 21}{9 - 7} = \frac{4}{2} = 2}\)

3. SIMPLIFY to find the y-intercept

• Use point \((7, 21)\) in equation \(\mathrm{y = mx + b}\)
• Substitute: \(\mathrm{21 = 2(7) + b}\)
• Solve: \(\mathrm{21 = 14 + b}\), so \(\mathrm{b = 7}\)

4. INFER the final equation

• Combine: \(\mathrm{h(x) = 2x + 7}\)

Answer: B. \(\mathrm{h(x) = 2x + 7}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors in the slope calculation, such as getting \(\mathrm{\frac{25-21}{9-7} = \frac{4}{2} = 1}\) instead of 2, or mix up the order of coordinates.

With wrong slope \(\mathrm{m = 1}\), they'd get \(\mathrm{y - 21 = 1(x - 7)}\), leading to \(\mathrm{y = x + 14}\). This doesn't match any answer choice exactly, leading to confusion and guessing.

Second Most Common Error:

Conceptual confusion about coordinate usage: Students directly use coordinate values as slope and y-intercept without calculation. They might see point \((7, 21)\) and think "slope = 7, y-intercept = 21."

This may lead them to select Choice C (\(\mathrm{h(x) = 7x + 21}\)).

The Bottom Line:

The key challenge is systematically applying the slope formula and then correctly using point substitution - students often rush and skip steps or make sign errors in the algebra.