A line passes through the points \((1, -30)\), \((3, 6)\), and \((5, 42)\). The equation of this line can be...

1. TRANSLATE the problem information

• Given information:
  • Three points on a line: \((1, -30)\), \((3, 6)\), and \((5, 42)\)
  • Need equation in form \(y = mx + k\)
  • Need to find \(m + k\)

• What this tells us: We have more points than we need (only need 2 to determine a unique line), which means the third point can serve as verification.

2. INFER the solution strategy

• Since we need to find both \(m\) and \(k\), we need two equations
• Each point gives us one equation when substituted into \(y = mx + k\)
• Strategy: Use any two points to create a system, then solve for \(m\) and \(k\)

3. TRANSLATE two points into equations

Using points \((1, -30)\) and \((3, 6)\):

• Point \((1, -30)\): \(-30 = m(1) + k\) → \(m + k = -30\)
• Point \((3, 6)\): \(6 = m(3) + k\) → \(3m + k = 6\)

4. SIMPLIFY the system of equations

Our system:

• \(m + k = -30\) (equation 1)
• \(3m + k = 6\) (equation 2)

Subtract equation 1 from equation 2:

\((3m + k) - (m + k) = 6 - (-30)\)

\(3m + k - m - k = 36\)

\(2m = 36\)

\(m = 18\)

5. SIMPLIFY to find k

Substitute \(m = 18\) into equation 1:

\(18 + k = -30\)

\(k = -48\)

6. APPLY CONSTRAINTS by verifying with the third point

Check with \((5, 42)\): \(y = 18(5) + (-48) = 90 - 48 = 42\) ✓

Therefore: \(m + k = 18 + (-48) = -30\)

Answer: (B) -30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly substitute coordinates into the linear equation form, often switching x and y values or making sign errors when dealing with negative coordinates.

For example, they might write: \(-30 = m + k(1)\) instead of \(-30 = m(1) + k\), treating \(k\) as the coefficient rather than \(m\). This fundamental misunderstanding of the linear equation structure leads to an incorrect system and wrong values for \(m\) and \(k\).

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly set up the system but make algebraic errors when solving it, such as incorrectly combining like terms or making sign errors during elimination.

For instance, they might get \(2m = -36\) instead of \(2m = 36\) when subtracting the equations, leading to \(m = -18\) and subsequently \(k = -12\), giving \(m + k = -30\). Wait, that's still the right answer...

Actually, a more likely error would be: \(2m = 6\) instead of \(2m = 36\), leading to \(m = 3\) and \(k = -33\), so \(m + k = -30\). That's still correct...

Let me reconsider: They might subtract incorrectly and get \(2m = 6 - (-30) = 6 + 30 = 36\), but then solve \(2m = 36\) as \(m = 36/2 = 18\) correctly, but then substitute incorrectly to get \(k\) wrong.

Actually, a more realistic error: They might get the arithmetic wrong when subtracting: \(6 - (-30) = 6 - 30 = -24\) instead of \(6 + 30 = 36\), giving them \(2m = -24\), so \(m = -12\), and then \(k = -30 - (-12) = -18\), making \(m + k = -30\). This still gives the right answer.

Let me think of a different error: They solve correctly but then calculate \(m + k = 18 + 48 = 66\) by forgetting the negative sign on \(k\).

This may lead them to select Choice (C) (66).

The Bottom Line:

The key challenge is managing multiple algebraic steps while keeping track of signs and variables. Success requires both systematic equation setup and careful algebraic manipulation.