Linear relationships and simultaneous equations

Lens

MotivationH

When do the thermometers agree?

Most of the world measures temperature in degrees Celsius; a few countries, including the United States, use degrees Fahrenheit. Water freezes at $0\,^\circ\mathrm{C}$ , which is $32\,^\circ\mathrm{F}$ , and boils (at sea level) at $100\,^\circ\mathrm{C}$ , which is $212\,^\circ\mathrm{F}$ . A warm afternoon in Accra at $25\,^\circ\mathrm{C}$ is $77\,^\circ\mathrm{F}$ .

Celsius $C$	$0$	$25$	$100$
Fahrenheit $F$	$32$	$77$	$212$

Here is a puzzle: is there a temperature at which both thermometers show the same number? At $0\,^\circ\mathrm{C}$ the Fahrenheit number is bigger by $32$ ; at $100\,^\circ\mathrm{C}$ , bigger by $112$ . The gap grows as it gets warmer — so perhaps it shrinks as it gets colder, and vanishes somewhere below zero?

More than 1,500 years ago, a Chinese textbook, the Sunzi Suanjing, posed a famous puzzle of the same kind: pheasants and rabbits share a cage; there are $35$ heads and $94$ feet. How many of each? Both puzzles ask for two unknown numbers satisfying two conditions at once. By the end of this lesson you will solve them in three different ways — and know exactly when such a puzzle has one answer, none, or infinitely many.

IntuitionH

Constant rate of change

A taxi in Lima charges a fixed $5$ soles plus $2$ soles per kilometre.

distance $x$ (km)	$0$	$1$	$2$	$3$	$4$
fare $y$ (soles)	$5$	$7$	$9$	$11$	$13$

Each extra kilometre adds the same amount, $2$ soles. The fare changes at a constant rate of $2$ soles per kilometre, and the formula is $y = 2x + 5$ . Plot the table and the points line up perfectly: a constant rate of change draws a straight line. The rate, $2$ , tells us how steep the line is; the starting value, $5$ , tells us where it crosses the vertical axis.

y = mx + b

Each step of 1 to the right changes y by exactly m.

y = mx + b

mm = 2bb = 5

Trace along xx = 1

y = mx + b: y = 7

DefinitionsH

Slope, linear functions and systems

Definition— slope

The slope of the line through two points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \neq x_2$ is

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}.

It is the change in $y$ per unit change in $x$ : a rate of change. A vertical line ( $x_1 = x_2$ ) has no slope — its run is $0$ , and we cannot divide by $0$ .

Definition— linear function

A linear function has the form $f(x) = mx + b$ for fixed numbers $m$ and $b$ (see functions). Its graph, drawn in the coordinate plane, is the line $y = mx + b$ , with slope $m$ and $y$ -intercept $b$ (the height where it crosses the $y$ -axis, at $x = 0$ ).

Definition— proportional relationship

A relationship is proportional if $y = kx$ for a constant $k$ , the constant of proportionality (see ratios and proportional reasoning). Proportional relationships are the linear functions with $b = 0$ : their graphs pass through the origin.

Definition— system of equations

A system of equations is a collection of equations in the same unknowns. A solution of the system is a choice of values that satisfies every equation at once. Equations solved together are also called simultaneous equations.

Worked examplesH

Worked examples

Example— Slopes from two points

Through $(2, 1)$ and $(6, 9)$ : $m = \dfrac{9 - 1}{6 - 2} = \dfrac{8}{4} = 2$ . The line rises $2$ for every $1$ to the right.
Through $(-1, 4)$ and $(3, -4)$ : $m = \dfrac{-4 - 4}{3 - (-1)} = \dfrac{-8}{4} = -2$ . A negative slope: the line falls.
Through $(1, 3)$ and $(5, 3)$ : $m = \dfrac{0}{4} = 0$ . A horizontal line: $y$ does not change.
Through $(3, 1)$ and $(3, 6)$ : the run is $0$ . This is the vertical line $x = 3$ ; its slope is undefined.

You may take the points in either order, as long as you subtract in the same order on top and bottom: $\dfrac{1 - 9}{2 - 6} = \dfrac{-8}{-4} = 2$ as well.

Example— A line from its slope and one point

Find the line with slope $3$ through $(2, 5)$ . It has the form $y = 3x + b$ , and the point must satisfy it: $5 = 3 \cdot 2 + b$ , so $b = -1$ . The line is $y = 3x - 1$ .

In general, the line with slope $m$ through $(x_1, y_1)$ is $y = mx + (y_1 - m x_1)$ , which can also be written in point–slope form: $y - y_1 = m(x - x_1)$ .

Example— A line through two points: Celsius to Fahrenheit

The points $(0, 32)$ and $(100, 212)$ give the slope $m = \dfrac{212 - 32}{100 - 0} = \dfrac{180}{100} = \dfrac{9}{5}$ , and the intercept is $32$ . So

F = \tfrac{9}{5}\,C + 32.

Check with the third point: $\tfrac{9}{5} \cdot 25 + 32 = 45 + 32 = 77$ . ✓ The slope says that each degree Celsius is $1.8$ degrees Fahrenheit.

Example— Recognising linear relationships in tables

In a table whose $x$ -values go up in equal steps, a relationship is linear exactly when the $y$ -values go up (or down) by equal steps too.

Rice at $60$ rupees per kilogram: $1 \mapsto 60$ , $2 \mapsto 120$ , $5 \mapsto 300$ . Linear and proportional: $y = 60x$ .
The taxi: $0 \mapsto 5$ , $1 \mapsto 7$ , $2 \mapsto 9$ . Linear but not proportional: doubling the distance does not double the fare ( $2$ km costs $9$ soles, not $2 \cdot 7 = 14$ ).
The area of a square: $1 \mapsto 1$ , $2 \mapsto 4$ , $3 \mapsto 9$ , $4 \mapsto 16$ . The steps are $3, 5, 7$ : not linear.

ProofH

Why a line has one slope

The examples suggest two facts: on a line, the slope does not depend on which two points we pick; and a constant rate of change always gives a formula $y = mx + b$ . Here are the proofs.

Theorem— the slope of y = mx + b is m

For any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line $y = mx + b$ with $x_1 \neq x_2$ , the slope between them is $m$ .

Proof

Since both points are on the line, $y_1 = m x_1 + b$ and $y_2 = m x_2 + b$ . So the rise is

y_2 - y_1 = (m x_2 + b) - (m x_1 + b) = m x_2 - m x_1 = m(x_2 - x_1),

and dividing by the run $x_2 - x_1 \neq 0$ gives exactly $m$ .

∎

Theorem— constant rate of change means linear

Suppose a function $f$ has a constant rate of change $m$ : for all $x_1$ and $x_2$ , $\;f(x_2) - f(x_1) = m(x_2 - x_1)$ . Then $f(x) = mx + b$ for every $x$ , where $b = f(0)$ .

Proof

Take $x_1 = 0$ and $x_2 = x$ : $f(x) - f(0) = m(x - 0) = mx$ , so $f(x) = mx + f(0)$ .

∎

Two slope triangles on the same line. The big one has run 5 and rise 3; the small one has run 2.5 and rise 1.5. They are similar triangles — the same shape at different sizes — so rise ÷ run is 3/5 for both.

The picture gives the geometric reason: slope triangles drawn on one line are similar (their angles match because the line has one direction), and similar triangles have proportional sides (see similarity). The algebraic proof above needs no picture at all.

Theorem— proportional means through the origin

The linear relationship $y = mx + b$ is proportional — that is, $mx + b = kx$ for all $x$ , for some constant $k$ — if and only if $b = 0$ .

Proof

If $mx + b = kx$ for every $x$ , put $x = 0$ : $b = 0$ . Conversely, if $b = 0$ then $y = mx$ , which is proportional with $k = m$ .

∎

Worked examplesH

Simultaneous equations: three methods

Each linear equation in two unknowns $x$ and $y$ , such as $x + y = 35$ , has infinitely many solutions — all the points of a line. A system of two such equations asks for the points on both lines: where they meet.

Example— Graphing: pheasants and rabbits

Let $x$ be the number of pheasants and $y$ the number of rabbits. Counting heads and feet (two feet per pheasant, four per rabbit):

x + y = 35, \qquad 2x + 4y = 94.

Solving each for $y$ : $y = 35 - x$ and $y = 23.5 - 0.5x$ . The lines meet at $(23, 12)$ .

Two conditions, one meeting point

Move the sliders and trace along the curves to read off values.

heads: x + y = 35
feet: 2x + 4y = 94

Trace along xx = 1

heads: x + y = 35: y = 34
feet: 2x + 4y = 94: y = 23

A graph shows the answer at a glance, but reading it is only as accurate as the drawing — a meeting point at $(23.4, 11.7)$ would look much the same. Algebra gives exact answers.

Example— Elimination: pheasants and rabbits again

Multiply the first equation by $2$ , giving $2x + 2y = 70$ , and subtract it from the second:

(2x + 4y) - (2x + 2y) = 94 - 70, \quad\text{so}\quad 2y = 24 \quad\text{and}\quad y = 12.

Then $x = 35 - 12 = 23$ . Check both equations: $23 + 12 = 35$ heads, and $2 \cdot 23 + 4 \cdot 12 = 46 + 48 = 94$ feet. ✓ Elimination works because, if both equations hold, any combination of them holds too.

Example— Substitution: when do the thermometers agree?

We want $F = \tfrac{9}{5}C + 32$ and $F = C$ at the same time. Substitute the second equation into the first:

C = \tfrac{9}{5}C + 32 \;\Longrightarrow\; -\tfrac{4}{5}C = 32 \;\Longrightarrow\; C = -40.

So $-40\,^\circ\mathrm{C} = -40\,^\circ\mathrm{F}$ — a temperature reached in the coldest winters of places such as Yakutsk in Siberia.

Example— No solution, and infinitely many

$x + y = 3$ and $2x + 2y = 10$ . Doubling the first gives $2x + 2y = 6$ , which contradicts $2x + 2y = 10$ . No solution: as $y = 3 - x$ and $y = 5 - x$ , the graphs are parallel lines.
$x + 2y = 4$ and $2x + 4y = 8$ . The second equation is just the first one doubled: both describe the same line. Infinitely many solutions: for every number $t$ , the point $(4 - 2t, t)$ works.

ProofH

One, none, or infinitely many

Theorem— where two lines meet

Consider the lines $y = m_1 x + b_1$ and $y = m_2 x + b_2$ .

If $m_1 \neq m_2$ , they meet in exactly one point, at $x = \dfrac{b_2 - b_1}{m_1 - m_2}$ .
If $m_1 = m_2$ and $b_1 \neq b_2$ , they never meet: they are parallel.
If $m_1 = m_2$ and $b_1 = b_2$ , they are the same line.

Proof

A meeting point needs $m_1 x + b_1 = m_2 x + b_2$ , that is, $(m_1 - m_2)x = b_2 - b_1$ . This is a linear equation in $x$ of the form $ax = b$ , and the lesson on linear equations showed it has exactly one solution if $m_1 - m_2 \neq 0$ , none if $m_1 = m_2$ and $b_2 - b_1 \neq 0$ , and every $x$ as a solution if both are zero. When there is exactly one $x$ , there is exactly one meeting point $(x, m_1 x + b_1)$ .

∎

Vertical lines $x = c$ are not of the form $y = mx + b$ , but they are easy: two different vertical lines are parallel, and a vertical line meets every non-vertical line exactly once. The next theorem covers all lines at once, vertical or not, and in fact all linear equations.

Theorem— two solutions give infinitely many

If two different points $(x_1, y_1)$ and $(x_2, y_2)$ both satisfy a linear equation $ax + by = e$ , then for every number $t$ so does the point

P_t = \big((1 - t)x_1 + t x_2,\; (1 - t)y_1 + t y_2\big),

and different values of $t$ give different points.

Proof

Substitute $P_t$ and regroup:

a\big((1 - t)x_1 + t x_2\big) + b\big((1 - t)y_1 + t y_2\big) = (1 - t)(a x_1 + b y_1) + t(a x_2 + b y_2) = (1 - t)e + te = e.

If $P_s = P_t$ with $s \neq t$ , then subtracting coordinates gives $(s - t)(x_2 - x_1) = 0$ and $(s - t)(y_2 - y_1) = 0$ , so $x_1 = x_2$ and $y_1 = y_2$ , contradicting that the points are different.

∎

The points $P_t$ make up the line through the two points ( $P_0$ is the first point, $P_1$ the second, $P_{1/2}$ the midpoint). Now apply the theorem to every equation of a system at once: if a system of linear equations — with any number of equations — has two different solutions, every $P_t$ solves all of them. So:

A system of linear equations has exactly one solution, no solution, or infinitely many. Never exactly two, or three.

Note

Going further: the general formula. For the system $ax + by = e$ , $cx + dy = f$ , multiply the first equation by $d$ , the second by $b$ , and subtract: the $y$ -terms cancel and $(ad - bc)\,x = ed - bf$ . Eliminating $x$ in the same way gives $(ad - bc)\,y = af - ec$ . So if the number $ad - bc$ is not zero, the system has exactly one solution,

x = \frac{ed - bf}{ad - bc}, \qquad y = \frac{af - ec}{ad - bc}.

The number $ad - bc$ is called the determinant of the system; when both equations describe genuine lines, it is zero exactly when the two lines are parallel or the same. You will meet it again in linear algebra.

Formal mathematicsF

Lines and systems in Lean

In Lean, slope is a small definition, and "the point $(x, y)$ is on the line $y = mx + b$ " is just the equation y = m * x + b. We work with the rational numbers Rat.

slopeLean 4Formally verified

/-- The slope of the line through `(x₁, y₁)` and `(x₂, y₂)`: rise over run,
`(y₂ − y₁) / (x₂ − x₁)`. It is meaningful only when `x₁ ≠ x₂`; the theorems below always
assume that. (Lean's `Rat` sets `a / 0 = 0`, which is a convention, not a slope.) -/
def slope (x₁ y₁ x₂ y₂ : Rat) : Rat := (y₂ - y₁) / (x₂ - x₁)