Straight Line

Mathematics · FE Reference Handbook section

Mathematics

8 formulas

10 exam-style examples

~60 min

All Mathematics lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Straight Line within Mathematics. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what straight line describes physically and when it applies.
State every one of the 8 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: radians vs degrees — set the calculator before the first trig entry.

Lecture

Why this section exists. Straight Line is the part of Mathematics that lets you connect an algebraic or calculus expression that must be evaluated exactly to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as short symbolic manipulations with one numeric evaluation at the end. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. radians vs degrees — set the calculator before the first trig entry. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Three engineers in hard hats and safety vests reviewing drawings on a truck tailgate. — Photo 1. Where this shows up in practice: straight line.
Capstone Studio instructional photograph

Mathematics — Straight Line: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes an algebraic or calculus expression that must be evaluated exactly. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 8 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

Ax + By + C	Quantity produced by "Ax + By + C = 0" — read its definition and unit from the handbook line directly above the equation.
y	Quantity produced by "y = mx + b," — read its definition and unit from the handbook line directly above the equation.
y – y1	Quantity produced by "y – y1 = m(x – x1)" — read its definition and unit from the handbook line directly above the equation.
m	Quantity produced by "m = (y2 – y1)/(x2 – x1)" — read its definition and unit from the handbook line directly above the equation.
α	Quantity produced by "α = arctan [(m2 – m1)/(1 + m2·m1)]" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

The general form of the equation is
The standard form of the equation is
which is also known as the slope-intercept form.
The point-slope form is
Given two points: slope,
The angle between lines with slopes m1 and m2 is
The distance between two points is
2 2

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Equation of a line through two points

A survey baseline passes through (2, 5) and (10, 21). What is the equation of the line, and what is y when x = 25?

Given

Point 1: (x₁, y₁) = (2, 5)
Point 2: (x₂, y₂) = (10, 21)

Find

y = mx + b, then y at x = 25

Start with the thinking

Two points fix a line uniquely: get the slope first, then use either point to get the intercept.
Keep the subtraction order the same in numerator and denominator or the slope sign flips.

Figure for Equation of a line through two points

Step-by-step solution

Slope
$m = (y_{2} - y_{1}) / (x_{2} - x_{1})$
Substituting
$m = (21 - 5) / (10 - 2) = 16/8 = 2.00$
Intercept from point 1
$b = y_{1} - m x_{1} = 5 - 2 (2) = 1.00$
Line
$y = 2 x + 1$
Evaluate at x = 25
$y = 2 (25) + 1 = 51.0$

Answer: y = 2x + 1, and y = 51.0 at x = 25

Why the other options are there

y = 0.5x + 4 (slope inverted)
y = 2x − 1 (intercept sign error)

Reference: FE Reference Handbook — Mathematics — Analytic geometry, straight line

Example 2

Perpendicular offset line

A property line has slope m = 2. A side lot line must be perpendicular to it and pass through (4, 3). What is its equation?

Given

m₁ = 2
Point on the new line: (4, 3)

Find

Equation of the perpendicular line

Start with the thinking

Perpendicular slopes are negative reciprocals.
Use point-slope form; do not re-derive the intercept from scratch.

Step-by-step solution

Perpendicular slope
$m_{2} = - 1/ m_{1} = - 1/2 = - 0.500$
Point-slope
$y - y_{1} = m_{2} (x - x_{1})$
Substituting
$y - 3 = - 0.5 (x - 4)$
Expand
$y = - 0.5 x + 2 + 3$
Result
$y = - 0.5 x + 5$

Answer: y = −0.5x + 5

Why the other options are there

y = 2x − 5 (same slope used)
y = −2x + 11 (reciprocal not taken)

Reference: FE Reference Handbook — Mathematics — Analytic geometry

Example 3

Straight line through two points: slope, intercept and length — Straight Line

A survey baseline passes through the points (-4, -1) and (3, 9). Find the slope of the straight line, its y-intercept, and the distance between the two points.

Given

P₁ = (-4, -1)
P₂ = (3, 9)

Find

Slope m, intercept b, and length d of the straight line

Start with the thinking

The slope of a straight line is the rise over the run between any two of its points.
Once m is known, the intercept follows from b = y − m x using either point.

Step-by-step solution

Formula
$m = (y_{2} - y_{1}) / (x_{2} - x_{1})$
Substituting
$m = (9 - - 1) / (3 - - 4) = 1.429$
Formula
$b = y_{1} - m x_{1}$
Substituting
$b = - 1 - (1.429) (- 4) = 4.714$
Formula — d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Substituting
$d = (7)^{2} + (10)^{2} = 12.207$
Line equation
$y = 1.429 x + 4.714$

Answer: m = 1.429, b = 4.714, d = 12.207 units

Why the other options are there

m = 0.700 (run over rise)
d = 17.00 (added instead of Pythagoras)

Reference: FE Reference Handbook — Mathematics → Straight Line

Example 4

Straight line through two points: slope, intercept and length — Straight Line (2)

A survey baseline passes through the points (-3, -2) and (4, 2). Find the slope of the straight line, its y-intercept, and the distance between the two points.

Given

P₁ = (-3, -2)
P₂ = (4, 2)

Find

Slope m, intercept b, and length d of the straight line

Start with the thinking

The slope of a straight line is the rise over the run between any two of its points.
Once m is known, the intercept follows from b = y − m x using either point.

Step-by-step solution

Formula
$m = (y_{2} - y_{1}) / (x_{2} - x_{1})$
Substituting
$m = (2 - - 2) / (4 - - 3) = 0.571$
Formula
$b = y_{1} - m x_{1}$
Substituting
$b = - 2 - (0.571) (- 3) = - 0.286$
Formula — d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Substituting
$d = (7)^{2} + (4)^{2} = 8.062$
Line equation
$y = 0.571 x + - 0.286$

Answer: m = 0.571, b = -0.286, d = 8.062 units

Why the other options are there

m = 1.750 (run over rise)
d = 11.00 (added instead of Pythagoras)

Reference: FE Reference Handbook — Mathematics → Straight Line

Example 5

Straight line through two points: slope, intercept and length — Straight Line (3)

A survey baseline passes through the points (-1, 2) and (1, -3). Find the slope of the straight line, its y-intercept, and the distance between the two points.

Given

P₁ = (-1, 2)
P₂ = (1, -3)

Find

Slope m, intercept b, and length d of the straight line

Start with the thinking

The slope of a straight line is the rise over the run between any two of its points.
Once m is known, the intercept follows from b = y − m x using either point.

Step-by-step solution

Formula
$m = (y_{2} - y_{1}) / (x_{2} - x_{1})$
Substituting
$m = (- 3 - 2) / (1 - - 1) = - 2.500$
Formula
$b = y_{1} - m x_{1}$
Substituting
$b = 2 - (- 2.500) (- 1) = - 0.500$
Formula — d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Substituting
$d = (2)^{2} + (- 5)^{2} = 5.385$
Line equation
$y = - 2.500 x + - 0.500$

Answer: m = -2.500, b = -0.500, d = 5.385 units

Why the other options are there

m = -0.400 (run over rise)
d = 7.00 (added instead of Pythagoras)

Reference: FE Reference Handbook — Mathematics → Straight Line

Example 6

Straight line through two points: slope, intercept and length — Straight Line (4)

A survey baseline passes through the points (-2, 4) and (6, 4). Find the slope of the straight line, its y-intercept, and the distance between the two points.

Given

P₁ = (-2, 4)
P₂ = (6, 4)

Find

Slope m, intercept b, and length d of the straight line

Start with the thinking

The slope of a straight line is the rise over the run between any two of its points.
Once m is known, the intercept follows from b = y − m x using either point.

Step-by-step solution

Formula
$m = (y_{2} - y_{1}) / (x_{2} - x_{1})$
Substituting
$m = (4 - 4) / (6 - - 2) = 0.000$
Formula
$b = y_{1} - m x_{1}$
Substituting
$b = 4 - (0.000) (- 2) = 4.000$
Formula — d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Substituting
$d = (8)^{2} + (0)^{2} = 8.000$
Line equation
$y = 0.000 x + 4.000$

Answer: m = 0.000, b = 4.000, d = 8.000 units

Why the other options are there

m = 8.000 (run over rise)
d = 8.00 (added instead of Pythagoras)

Reference: FE Reference Handbook — Mathematics → Straight Line

Example 7

Straight line through two points: slope, intercept and length — Straight Line (5)

A survey baseline passes through the points (3, -2) and (7, -3). Find the slope of the straight line, its y-intercept, and the distance between the two points.

Given

P₁ = (3, -2)
P₂ = (7, -3)

Find

Slope m, intercept b, and length d of the straight line

Start with the thinking

The slope of a straight line is the rise over the run between any two of its points.
Once m is known, the intercept follows from b = y − m x using either point.

Step-by-step solution

Formula
$m = (y_{2} - y_{1}) / (x_{2} - x_{1})$
Substituting
$m = (- 3 - - 2) / (7 - 3) = - 0.250$
Formula
$b = y_{1} - m x_{1}$
Substituting
$b = - 2 - (- 0.250) (3) = - 1.250$
Formula — d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Substituting
$d = (4)^{2} + (- 1)^{2} = 4.123$
Line equation
$y = - 0.250 x + - 1.250$

Answer: m = -0.250, b = -1.250, d = 4.123 units

Why the other options are there

m = -4.000 (run over rise)
d = 5.00 (added instead of Pythagoras)

Reference: FE Reference Handbook — Mathematics → Straight Line

Example 8

Straight line through two points: slope, intercept and length — Straight Line (6)

A survey baseline passes through the points (-5, 2) and (-3, 4). Find the slope of the straight line, its y-intercept, and the distance between the two points.

Given

P₁ = (-5, 2)
P₂ = (-3, 4)

Find

Slope m, intercept b, and length d of the straight line

Start with the thinking

The slope of a straight line is the rise over the run between any two of its points.
Once m is known, the intercept follows from b = y − m x using either point.

Step-by-step solution

Formula
$m = (y_{2} - y_{1}) / (x_{2} - x_{1})$
Substituting
$m = (4 - 2) / (- 3 - - 5) = 1.000$
Formula
$b = y_{1} - m x_{1}$
Substituting
$b = 2 - (1.000) (- 5) = 7.000$
Formula — d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Substituting
$d = (2)^{2} + (2)^{2} = 2.828$
Line equation
$y = 1.000 x + 7.000$

Answer: m = 1.000, b = 7.000, d = 2.828 units

Why the other options are there

m = 1.000 (run over rise)
d = 4.00 (added instead of Pythagoras)

Reference: FE Reference Handbook — Mathematics → Straight Line

Example 9

Straight line through two points: slope, intercept and length — Straight Line (7)

A survey baseline passes through the points (-3, 0) and (2, -1). Find the slope of the straight line, its y-intercept, and the distance between the two points.

Given

P₁ = (-3, 0)
P₂ = (2, -1)

Find

Slope m, intercept b, and length d of the straight line

Start with the thinking

The slope of a straight line is the rise over the run between any two of its points.
Once m is known, the intercept follows from b = y − m x using either point.

Step-by-step solution

Formula
$m = (y_{2} - y_{1}) / (x_{2} - x_{1})$
Substituting
$m = (- 1 - 0) / (2 - - 3) = - 0.200$
Formula
$b = y_{1} - m x_{1}$
Substituting
$b = 0 - (- 0.200) (- 3) = - 0.600$
Formula — d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Substituting
$d = (5)^{2} + (- 1)^{2} = 5.099$
Line equation
$y = - 0.200 x + - 0.600$

Answer: m = -0.200, b = -0.600, d = 5.099 units

Why the other options are there

m = -5.000 (run over rise)
d = 6.00 (added instead of Pythagoras)

Reference: FE Reference Handbook — Mathematics → Straight Line

Example 10

Straight line through two points: slope, intercept and length — Straight Line (8)

A survey baseline passes through the points (-3, -6) and (-1, 9). Find the slope of the straight line, its y-intercept, and the distance between the two points.

Given

P₁ = (-3, -6)
P₂ = (-1, 9)

Find

Slope m, intercept b, and length d of the straight line

Start with the thinking

The slope of a straight line is the rise over the run between any two of its points.
Once m is known, the intercept follows from b = y − m x using either point.

Step-by-step solution

Formula
$m = (y_{2} - y_{1}) / (x_{2} - x_{1})$
Substituting
$m = (9 - - 6) / (- 1 - - 3) = 7.500$
Formula
$b = y_{1} - m x_{1}$
Substituting
$b = - 6 - (7.500) (- 3) = 16.500$
Formula — d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Substituting
$d = (2)^{2} + (15)^{2} = 15.133$
Line equation
$y = 7.500 x + 16.500$

Answer: m = 7.500, b = 16.500, d = 15.133 units

Why the other options are there

m = 0.133 (run over rise)
d = 17.00 (added instead of Pythagoras)

Reference: FE Reference Handbook — Mathematics → Straight Line

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given an algebraic or calculus expression that must be evaluated exactly, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Straight Line contains 8 relations; you must be able to find this page in under 15 seconds.
Exam style: short symbolic manipulations with one numeric evaluation at the end.
Unit rule: radians vs degrees — set the calculator before the first trig entry.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

radians vs degrees — set the calculator before the first trig entry
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.

Straight Line

Learning objectives

Lecture

Theory, developed

Notation used in this section

Handbook notes for this section

Core formulas for this FE topic

Ax+By+C=0

y=mx+b,

y−y1​=m(x−x1​)

m=(y2​−y1​)/(x2​−x1​)

α = arctan [(m2 – m1)/(1 + m2·m1)]Needs technical review

Twolinesareperpendicularifm1​=−1/m2​

_ y2 - y1i + ^ x2 - x1hNeeds technical review

d=Needs technical review

Worked exam-style examples

Self-check

Chapter summary

Common traps in this section