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Mathematics I · Week 2 — Coordinate system & straight lines

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Mathematics I · Week 2 — Coordinate system & straight lines

1400 words

7 min read

2026-08-16

Reading compass

Now · Week map

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.

Part of the Quiz 2 prep system · Weeks 1–8 index · Pattern atlas · Formula chains.

Week map

Points → slope → line forms → parallel/perpendicular → applications

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).

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

Open interactive formula desk · Week 2 tab.

Deep study

Mathematics I · Week 2 — Lines and slope

Coordinate plane tools for reading graphs, writing equations, and checking parallelism.

Week map

Points and distance → slope as rate of change → point-slope and slope-intercept → general form → parallel and perpendicular → intercepts and applications.

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).

Mini-example: Through

(1,4)

and

(5,10)

m = \frac{10-4}{5-1} = \frac{6}{4} = \frac{3}{2}

. Rise 3 per run 2.

Line equations

Point-slope form

y - y_1 = m(x - x_1)

— use when point and slope known.

Through

(2,-1)

with

m = -3

y + 1 = -3(x - 2)

→

y = -3x + 5

Slope-intercept form

y = mx + c

—

c

is $y$ -intercept (value when

x=0

General form

ax + by + c = 0

— useful for parallel/perpendicular via coefficients; slope

m = -\frac{a}{b}

when

b \neq 0

Trap: In

3x + 2y - 8 = 0

, the

y

-intercept is not

-8

; solve:

2y = -3x + 8

y = -\frac{3}{2}x + 4

, intercept

4

Parallel and perpendicular

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

Mini-example: Line with

m = \frac{2}{5}

. Perpendicular slope is

-\frac{5}{2}

. Parallel slope stays

\frac{2}{5}

Vertical ⊥ horizontal always. Two vertical lines are parallel; vertical ⊥ horizontal with undefined vs 0 slope — do not apply

m_1 m_2 = -1

blindly.

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 $(x_2,y_2)$ : $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ .

Mini-example: Line

y = 2x - 6

x

-intercept:

0 = 2x - 6

x = 3

. Point

(3,0)

y

-intercept:

(0,-6)

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

Example 1 — Slope and intercept. Points

(-2,5)

and

(4,1)

m = \frac{1-5}{4-(-2)} = \frac{-4}{6} = -\frac{2}{3}

. Using point

(4,1)

y - 1 = -\frac{2}{3}(x - 4)

Example 2 — Parallel line. Given

2x - 5y + 10 = 0

, slope

m = \frac{2}{5}

. Line through

(0,3)

parallel:

y - 3 = \frac{2}{5}x

→

y = \frac{2}{5}x + 3

Example 3 — Perpendicular. Slope of

y = 4x + 1

4

. Perpendicular slope

-\frac{1}{4}

. Through

(8,2)

y - 2 = -\frac{1}{4}(x - 8)

Example 4 — Distance.

(1,2)

(4,6)

d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9+16} = 5

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)

Find the slope of the line through $(5,-1)$ and $(-3,7)$ .
Write the equation of the line with slope $-\frac{3}{4}$ passing through $(12,5)$ in slope-intercept form.
Line $L$ has equation $3x + y = 9$ . Write an equation for the line through $(2,1)$ perpendicular to $L$ .
Where does the line $4x - 2y = 12$ cross the $x$ -axis and the $y$ -axis?
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

Item	Formula
Slope	$m = \frac{y_2 - y_1}{x_2 - x_1}$
Distance	$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Parallel	$m_1 = m_2$
Perpendicular	$m_1 m_2 = -1$

Pattern families

Easy — Slope from two points

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

Medium — Line through point with condition

Start from point-slope form when a point and slope are known. For parallel to a given line, copy its slope; for perpendicular, use the negative reciprocal. Convert to the form requested (slope-intercept or general).

Hard — Geometry with lines

Combine distance formula with line equations to find intersection or verify perpendicularity via slopes. Sketch roughly to catch sign errors. Check feasibility: perpendicular slope to a horizontal line is undefined (vertical).

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

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

Practice loop

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