Week 2 — coordinate system & straight lines

Quiz 2 scope: Weeks 1–8 per IITM May 2026 foundation courses. Source baseline: IITM BS admissions important-dates calendar · May 2026 cycle. Times on assessments are operational conventions — verify hall ticket.

Week map

Classify → Represent → Execute → Trap-check

• Recognize: - Two points → find line.
• Procedure: Subtract coordinates in the same order for both numerator and denominator. Simplify the fraction and state whether the line rises, falls, or is constant. If $x_1 = x_2$, report undefined slope and write $x = k$.
• Variations / traps: Watch for: Mixing up $(y_2-y_1)/(x_2-x_1)$ with reversed differences inconsistently.

Formula chain (compressed)

Δx, Δy → slope m → line forms → parallel (same m) vs perpendicular (m₁m₂=−1).

1. Change — Δx = x₂−x₁, Δy = y₂−y₁ — before plugging into formulas
2. Slope — m = Δy/Δx — two distinct points
3. Point-slope — y−y₁ = m(x−x₁) — point + slope known
4. Slope-intercept — y = mx + c — c is y-intercept at x=0
5. Perpendicular — m₂ = −1/m₁ — non-vertical lines

Deep study

Mathematics I · Week 2 — Lines and slope

Week map

Coordinate notation

• $(x,y)$ → ordered pair → horizontal then vertical → $(3,-2)$ is 3 right, 2 down.
• $m$ → slope → rise over run → $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• $\Delta y$ → change in $y$ → vertical difference → if $y$ goes from 4 to 7, $\Delta y = 3$.
• $\Delta x$ → change in $x$ → horizontal difference → keep subtraction order consistent with $y$.

Reading slope

• $m > 0$ → line rises left to right.
• $m < 0$ → line falls.
• $m = 0$ → horizontal; equation $y = k$.
• Undefined slope → vertical; equation $x = k$ (not $y = mx + b$ form).

Line equations

Point-slope form

Slope-intercept form

General form

Parallel and perpendicular

• Parallel: same slope $m_1 = m_2$ (including both vertical).
• Perpendicular (neither vertical): $m_1 m_2 = -1$ → $m_2 = -\frac{1}{m_1}$.

Intercepts and distance

• $x$-intercept: set $y=0$, solve for $x$.
• $y$-intercept: set $x=0$, solve for $y$.
• Distance between $(x_1,y_1)$ and $(x_2,y_2)$: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.

Pattern families

Easy — Slope from two points

• Compute $m$ with consistent $\Delta y / \Delta x$.
• State rise/run interpretation in words.
• Identify horizontal/vertical from equal $x$ or equal $y$ coordinates.

Medium — Equation from conditions

• Line through one point with given slope → point-slope, simplify.
• Line parallel/perpendicular to known line through a point → copy or negate-reciprocal slope first.
• Convert between forms as requested.

Hard — Geometry with lines

• Find intersection by solving linear system (substitution).
• Verify perpendicularity via slopes after finding both lines.
• Distance from point to point combined with intercept puzzles.

Worked mini-examples

Traps

• Inconsistent difference order: $\frac{y_2-y_1}{x_2-x_1}$ not mixed with $\frac{y_1-y_2}{x_2-x_1}$ wrongly.
• Using $m_1 m_2 = -1$ when one line is horizontal.
• Confusing general-form constant with intercept.
• Assuming lines with different slopes must intersect off-screen — they intersect somewhere unless parallel.

Diagnostic (try yourself)

1. Find the slope of the line through $(5,-1)$ and $(-3,7)$.
2. Write the equation of the line with slope $-\frac{3}{4}$ passing through $(12,5)$ in slope-intercept form.
3. Line $L$ has equation $3x + y = 9$. Write an equation for the line through $(2,1)$ perpendicular to $L$.
4. Where does the line $4x - 2y = 12$ cross the $x$-axis and the $y$-axis?
5. Are the lines $y = 5x - 2$ and $5x - y = 7$ parallel, perpendicular, or neither?

ChatGPT prep archive

• Chatty reference archive

Archived import for extra depth — complements the notes above, not official IITM material.

Core concepts

• Slope as rise over run; undefined slope for vertical lines; zero slope for horizontal.
• Forms: point-slope $y - y_1 = m(x - x_1)$, slope-intercept $y = mx + c$, general $ax + by + c = 0$.
• Parallel lines share slope; perpendicular slopes multiply to $-1$ (when neither is vertical).
• Intercepts: $x$-intercept where $y=0$; $y$-intercept where $x=0$.

Notation & vocabulary

Pattern families

Easy — Slope from two points

Medium — Line through point with condition

Hard — Geometry with lines

Drill these on the pattern atlas — filter to week 2.

Traps

• Mixing up $(y_2-y_1)/(x_2-x_1)$ with reversed differences inconsistently.
• Using $m_1 m_2 = -1$ when one line is horizontal and the other should be vertical.
• Forgetting that parallel vertical lines both have undefined slope.
• Confusing intercept $c$ in $y=mx+c$ with constant term in $ax+by+c=0$.

Retrieval prompts

1. What slope characterizes a line perpendicular to $y = 3x - 2$?
2. How do you write the equation of a vertical line through $(4,-1)$?
3. When are two lines parallel in slope-intercept form?

Practice loop

1. Read Deep study (if present) or core concepts once.
2. Recite the formula chain without looking.
3. Open one easy pattern on the interactive atlas for week 2.
4. Attempt without solutions; mark studied after an honest try.
5. Say one trap aloud before closing the tab.

Related

• Full weeks 1–8 prep guide
• Pattern atlas note
• Formula chain desk