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# 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.

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?