The table shows two values of x and their corresponding values of y. The graph of the linear equation representing...

1. TRANSLATE the problem information

• Given information:
  • Two points on a linear relationship: \((-12, -45)\) and \((6, 45)\)
  • The line passes through point \((1/4, a)\) where a is unknown
• What this tells us: We need to find the linear equation first, then use it to find the y-coordinate when \(\mathrm{x = 1/4}\)

2. INFER the solution strategy

• Since we have two points on a line, we can find the slope and then the equation
• Once we have the equation, we substitute \(\mathrm{x = 1/4}\) to find the value of a
• Strategy: Find slope → Write equation → Substitute \(\mathrm{x = 1/4}\)

3. SIMPLIFY to find the slope

• Using the slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{45 - (-45)}{6 - (-12)}}\)
• \(\mathrm{m = \frac{90}{18}}\)
• \(\mathrm{m = 5}\)

4. INFER and write the linear equation

• Using point-slope form with point \((6, 45)\): \(\mathrm{y - 45 = 5(x - 6)}\)
• SIMPLIFY: \(\mathrm{y - 45 = 5x - 30}\), so \(\mathrm{y = 5x + 15}\)

5. SIMPLIFY to find the value of a

• Substitute \(\mathrm{x = 1/4}\) into \(\mathrm{y = 5x + 15}\)
• \(\mathrm{a = 5(1/4) + 15}\)
• \(\mathrm{a = 5/4 + 15}\)
• Convert 15 to fourths: \(\mathrm{a = 5/4 + 60/4}\)
• \(\mathrm{a = 65/4}\)
• As a decimal (use calculator): \(\mathrm{a = 16.25}\)

Answer: 65/4, 16.25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when handling the fraction 1/4 in the final substitution step.

Many students correctly find the equation \(\mathrm{y = 5x + 15}\), but then struggle with \(\mathrm{a = 5(1/4) + 15}\). They might calculate \(\mathrm{5(1/4) = 5/4 = 1.25}\), but then add \(\mathrm{1.25 + 15 = 16.25}\) without converting to a common fraction form. While 16.25 is correct, they may not recognize that 65/4 is the exact fractional form, leading to confusion about which answer format to use.

Second Most Common Error:

Poor INFER reasoning: Students attempt to use the point \((1/4, a)\) as if it's a known point to help find the equation, rather than recognizing it's the target point they need to evaluate.

This backwards thinking leads them to try setting up three equations with three unknowns or getting confused about which points are given versus which point contains the unknown. This causes them to get stuck and guess rather than following the logical sequence of finding the equation first.

The Bottom Line:

This problem tests whether students can distinguish between given information (the table points) and target information (the point with the unknown), then execute a systematic approach: find the equation, then evaluate it at the target x-value.