The slope and the equation of a line

Geometry, grade 10 (5 units) Lesson 2 of 26Learning

The slope and the equation of a line

The slope of a line

The slope of a line is the change in $\displaystyle y$ for each step of $\displaystyle 1$ in $\displaystyle x$ . Between any two points of the line it is the change in $\displaystyle y$ divided by the change in $\displaystyle x$ :

\displaystyle m

\displaystyle =

\displaystyle \frac{y_2-y_1}{x_2-x_1}

\displaystyle x_1\neq x_2

\displaystyle y=\frac{2}{3}x+1

\displaystyle A=(0,1)

\displaystyle B=(6,5)

From $\displaystyle A(0,1)$ to $\displaystyle B(6,5)$ : $\displaystyle 6$ to the right and $\displaystyle 4$ up, so $\displaystyle {m}={\frac{4}{6}}={\frac{2}{3}}$ .
Any two points of a line give the same slope: the triangles of the steps are similar.
$\displaystyle m>0$ – the line rises from left to right; $\displaystyle m<0$ – it falls; $\displaystyle {m}={0}$ – it is parallel to the $\displaystyle x$ -axis.

Exercise 1

Find the slope of the line through $\displaystyle A(-1,4)$ and $\displaystyle B(3,-4)$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A line by its slope and a point

A point $\displaystyle (x,y)$ is on the line of slope $\displaystyle m$ through $\displaystyle (x_1,y_1)$ exactly when the slope between the two points is $\displaystyle m$ : $\displaystyle {\frac{y-y_1}{x-x_1}}={m}$ . Multiplied out, this is the equation of the line:

\displaystyle y-y_1

\displaystyle =

\displaystyle m(x-x_1)

Example: the line of slope $\displaystyle -2$ through $\displaystyle (3,1)$ .

$\displaystyle {y-1}={-2(x-3)}$ ;
$\displaystyle {y-1}={-2x+6}$ ;
$\displaystyle {y}={-2x+7}$ .

A line through two points: first the slope between them, then the equation by the slope and one of the points.

Exercise 2

Write the equation of the line of slope $\displaystyle 3$ through the point $\displaystyle P(2,-1)$ .

\displaystyle P=(2,-1)

Exercise 3

Write the equation of the line through $\displaystyle A(1,5)$ and $\displaystyle B(4,-1)$ .

\displaystyle A=(1,5)

\displaystyle B=(4,-1)

The explicit and the general equation

The same line can be written in two forms:

explicit – $\displaystyle {y}={mx+b}$ : the slope $\displaystyle m$ and the point $\displaystyle (0,b)$ on the $\displaystyle y$ -axis can be read at once, and for each $\displaystyle x$ the equation gives $\displaystyle y$ ;
general – $\displaystyle {Ax+By+C}={0}$ : every line has one, a line parallel to the $\displaystyle y$ -axis too; to read its slope, isolate $\displaystyle y$ .

Example: $\displaystyle {2x+3y-6}={0}$ .

$\displaystyle {3y}={-2x+6}$ , so $\displaystyle {y}={-\frac{2}{3}x+2}$ ;
the slope is $\displaystyle -\frac{2}{3}$ , and the line crosses the $\displaystyle y$ -axis at $\displaystyle (0,2)$ .

The general form is not unique: $\displaystyle {4x+6y-12}={0}$ is the same line – the equation times $\displaystyle 2$ .

A line parallel to the y-axis

All the points of a line parallel to the $\displaystyle y$ -axis have the same $\displaystyle x$ , so its equation is $\displaystyle {x}={k}$ .

\displaystyle x=3

\displaystyle y=2

$\displaystyle {x}={3}$ has no slope: between two of its points the change in $\displaystyle x$ is $\displaystyle 0$ , and you cannot divide by $\displaystyle 0$ .
It is not the graph of a function: a function gives one $\displaystyle y$ for each $\displaystyle x$ , and here $\displaystyle {x}={3}$ has every $\displaystyle y$ .
In the general form it is $\displaystyle {x-3}={0}$ : $\displaystyle {B}={0}$ .
The dashed line $\displaystyle {y}={2}$ is parallel to the $\displaystyle x$ -axis: its slope is $\displaystyle 0$ , and it is the graph of a function – the same $\displaystyle y$ for every $\displaystyle x$ .

Exercise 4

The general equation of a line is $\displaystyle {5x-2y+4}={0}$ .
Write its explicit equation.

Exercise 5

Write the equation of each line.
Which of the lines are graphs of functions?

The line through $\displaystyle (3,1)$ and $\displaystyle (3,5)$ .
The line through $\displaystyle (-2,4)$ and $\displaystyle (6,4)$ .
The line through the origin and $\displaystyle (2,-6)$ .

A line with a parameter

An equation with a letter besides $\displaystyle x$ and $\displaystyle y$ – a parameter – is a whole family of lines, one for each value of the parameter. A condition on the line is an equation in the parameter.

Example: $\displaystyle {(k-1)x+2y}={k}$ . For which $\displaystyle k$ does the line pass through $\displaystyle (3,1)$ ?

Substitute the point: $\displaystyle {3(k-1)+2\cdot1}={k}$ ;
$\displaystyle {3k-1}={k}$ , so $\displaystyle {k}={\frac{1}{2}}$ ;
the line: $\displaystyle {-\frac{1}{2}x+2y}={\frac{1}{2}}$ , or $\displaystyle {y}={\frac{1}{4}x+\frac{1}{4}}$ .

Exercise 6

The line: $\displaystyle {kx+(k+2)y}={12}$ .

For which

\displaystyle k

does the line pass through the point

\displaystyle (1,2)

For which $\displaystyle k$ is the line parallel to the $\displaystyle y$ -axis?
For which $\displaystyle k$ is it parallel to the $\displaystyle x$ -axis?
Write the line’s equation in each case.

Formulas: geometry, grade 10 (5 units)

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Introduction to analytic geometry

The distance between the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{(x_1-x_2)^{2}+(y_1-y_2)^{2}}$
The midpoint of the segment
$\displaystyle M\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)$
The slope of a line through two points
$\displaystyle m$ $\displaystyle =$ $\displaystyle \frac{y_2-y_1}{x_2-x_1}$ $\displaystyle x_1\neq x_2$
The line of slope $\displaystyle m$ through the point $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1$ $\displaystyle =$ $\displaystyle m(x-x_1)$
The explicit equation of a line – $\displaystyle m$ the slope, $\displaystyle b$ where it crosses the $\displaystyle y$ -axis
$\displaystyle y=mx+b$
The general equation of a line
$\displaystyle Ax+By+C=0$
Parallel lines
$\displaystyle m_1$ $\displaystyle =$ $\displaystyle m_2$
Perpendicular lines
$\displaystyle m_1\cdot m_2$ $\displaystyle =$ $\displaystyle -1$
A canonical circle – its centre at the origin, $\displaystyle R$ its radius
$\displaystyle x^{2}+y^{2}=R^{2}$
The circle with centre $\displaystyle (a,b)$ and radius $\displaystyle R$
$\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}$

Midsegments and the meeting point of the medians

A triangle’s midsegment – $\displaystyle DE$ is parallel to $\displaystyle BC$
$\displaystyle DE$ $\displaystyle =$ $\displaystyle \frac{1}{2}BC$
A trapezoid’s midsegment – $\displaystyle a$ , $\displaystyle b$ the bases
$\displaystyle EF$ $\displaystyle =$ $\displaystyle \frac{a+b}{2}$
The meeting point of the medians $\displaystyle G$ divides the median $\displaystyle AD$ in the ratio $\displaystyle 2:1$
$\displaystyle AG$ $\displaystyle =$ $\displaystyle 2GD$
Pythagoras’ theorem and its converse: the triangle is right-angled, the right angle opposite $\displaystyle c$
$\displaystyle a^{2}+b^{2}=c^{2}$

Proportion and similarity

A property of a proportion
$\displaystyle \frac{a}{b}=\frac{c}{d}\ \Rightarrow\ \frac{a+b}{b}=\frac{c+d}{d}$
Thales’ theorem – $\displaystyle DE\parallel BC$
$\displaystyle \frac{AD}{DB}$ $\displaystyle =$ $\displaystyle \frac{AE}{EC}$
The extended Thales theorem – $\displaystyle DE\parallel BC$
$\displaystyle \frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC}$
The angle bisector theorem – $\displaystyle AD$ bisects the angle $\displaystyle A$
$\displaystyle \frac{BD}{DC}$ $\displaystyle =$ $\displaystyle \frac{AB}{AC}$
The similarity ratio – corresponding sides
$\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=k$
Heights, medians, perimeters, the radii of the circumscribed and of the inscribed circles
$\displaystyle \frac{h_1}{h_2}=\frac{m_1}{m_2}=\frac{P_1}{P_2}=\frac{R_1}{R_2}=\frac{r_1}{r_2}=k$
The ratio of the areas
$\displaystyle \frac{S_1}{S_2}$ $\displaystyle =$ $\displaystyle k^{2}$

The circle

An inscribed angle is half the central angle on the same arc
$\displaystyle \angle ACB$ $\displaystyle =$ $\displaystyle \frac{1}{2}\angle AOB$
An inscribed angle on the diameter $\displaystyle AB$
$\displaystyle \angle ACB=90^{\circ}$
A quadrilateral $\displaystyle ABCD$ inscribed in a circle – opposite angles
$\displaystyle \angle A+\angle C$ $\displaystyle =$ $\displaystyle 180^{\circ}$
The angle between the tangent $\displaystyle AT$ and the chord $\displaystyle AB$ equals an inscribed angle on $\displaystyle AB$ from its other side
$\displaystyle \angle BAT$ $\displaystyle =$ $\displaystyle \angle ACB$
Two tangents from the point $\displaystyle P$
$\displaystyle PA=PB$
A quadrilateral $\displaystyle ABCD$ circumscribing a circle
$\displaystyle AB+CD$ $\displaystyle =$ $\displaystyle BC+AD$
The chords $\displaystyle AB$ and $\displaystyle CD$ meet at the point $\displaystyle E$
$\displaystyle AE\cdot EB$ $\displaystyle =$ $\displaystyle CE\cdot ED$
Two secants from the point $\displaystyle P$
$\displaystyle PA\cdot PB$ $\displaystyle =$ $\displaystyle PC\cdot PD$
A tangent $\displaystyle PT$ and a secant $\displaystyle PAB$ from the point $\displaystyle P$
$\displaystyle PT^{2}$ $\displaystyle =$ $\displaystyle PA\cdot PB$