A line in the xy-plane passes through the points \((2, 3)\) and \((4, 1)\). What is the y-intercept of this...

1. TRANSLATE the problem information

• Given information:
  • Two points on the line: \(\mathrm{(2, 3)}\) and \(\mathrm{(4, 1)}\)
  • Need to find: y-intercept of this line
• The y-intercept is where the line crosses the y-axis (when \(\mathrm{x = 0}\))

2. INFER the approach

• To find the y-intercept, we need the equation of the line first
• Strategy: Find slope → Write equation → Find y-intercept
• We'll use the slope formula, then point-slope form

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = (y_2 - y_1)/(x_2 - x_1)}\)
• \(\mathrm{m = (1 - 3)/(4 - 2)}\)
  \(\mathrm{= -2/2}\)
  \(\mathrm{= -1}\)
• The slope is -1

4. SIMPLIFY to find the equation using point-slope form

• Using point \(\mathrm{(2, 3)}\): \(\mathrm{y - 3 = -1(x - 2)}\)
• Expand: \(\mathrm{y - 3 = -x + 2}\)
• Add 3 to both sides: \(\mathrm{y = -x + 2 + 3}\)
• Final equation: \(\mathrm{y = -x + 5}\)

5. INFER how to find the y-intercept

• The y-intercept occurs when \(\mathrm{x = 0}\)
• Substitute \(\mathrm{x = 0}\) into our equation: \(\mathrm{y = -(0) + 5 = 5}\)
• The y-intercept is the point \(\mathrm{(0, 5)}\)

Answer: B. \(\mathrm{(0, 5)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing y-intercept with x-intercept

Students may correctly find the equation \(\mathrm{y = -x + 5}\), but then think the intercept is asking for when \(\mathrm{y = 0}\). Setting \(\mathrm{-x + 5 = 0}\) gives \(\mathrm{x = 5}\), leading them to think the answer is \(\mathrm{(5, 0)}\).

This may lead them to select Choice D. \(\mathrm{(5, 0)}\)

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors in the point-slope form

When expanding \(\mathrm{y - 3 = -1(x - 2)}\), students might incorrectly write \(\mathrm{y - 3 = -x - 2}\) instead of \(\mathrm{y - 3 = -x + 2}\). This leads to \(\mathrm{y = -x + 1}\), and a y-intercept of \(\mathrm{(0, 1)}\). Since this isn't an option, it causes confusion and guessing.

This leads to confusion and guessing among the available choices.

The Bottom Line:

This problem tests both computational skills (slope and equation manipulation) and conceptual understanding (what y-intercept means). The key is remembering that y-intercept means "where \(\mathrm{x = 0}\)" and carefully tracking signs during algebraic manipulation.