Equation of a line in the standard form

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Learning objectives

You will develop the following competences: communicating in English, mathematical, IT and basic scientific and technical competence, your learning skills.
You will gett to know the equation of a straight line in the standard form.
You will convert from the standard form to the slope‑intercept form.

Learning effect

You gett to know equation of a straight line in the standard form.
You convert from the standard form to the slope‑intercept form.

Recollect the information about the slope‑intercept equation of a lineequation of a lineequation of a line. The aim of the class is getting to know the standard equation of the straight linelineline. Think – can any linelineline in the coordinate systemcoordinate systemcoordinate system be described with the equation in the slope‑intercept form? Check your hypotheses and formulate the conclusion.

The conclusion: A linelineline perpendicular to OX axis cannot be described with the slope‑intercept equation.

Find in any available sources information about other equations that can be used to describe lines in the coordinate systemcoordinate systemcoordinate system. Write down the definition of the standard equation of a lineequation of a lineequation of a line.

standard equation of a line

Definition: standard equation of a line

The standard equation of a lineequation of a lineequation of a line is denoted as follows $A x + B y + C = 0$ , where $A^{2} + B^{2} \neq 0$

Condition $A^{2} + B^{2} \neq 0$ means that coefficients $A$ and $B$ cannot equal zero at the same time.

Analyse the Interactive Illustration which shows the method of converting from the slope‑intercept equation of a lineequation of a lineequation of a line to the standard equation and back again. Formulate your conclusions.

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Converting from the slope-intercept equation of a line to the standard equation and back again.

Source: GroMar, licencja: CC BY 3.0.

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nagranie abstraktu

In order to convert from the slope‑intercept equation of a lineequation of a lineequation of a line to the standard equation, you need to move all terms of the equation onto one side and multiply both sides of the equation by the same number, so that all the obtained coefficients are integers.
In order to convert from the standard equation of a lineequation of a lineequation of a line $A x + B y + C = 0$ to the slope‑intercept equation, you need to move terms $A x$ and $C$ onto the right side of the equation. If $B \neq 0$ , you need to divide both sides of the equation by $B$ .

Work out how to define the standard equation of a lineequation of a lineequation of a line intercepting at two points.

Task 1

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nagranie abstraktu

Define the standard equation of a lineequation of a lineequation of a line intercepting at two points $A (x_{A}; y_{A})$ and $B (x_{B}; y_{B})$ , where $x_{A} \neq x_{B}$ . Formulate your conclusion.

The standard equation of a lineequation of a lineequation of a line intercepting at two points $A (x_{A}; y_{A})$ and $B (x_{B}; y_{B})$ , where $x_{A} \neq x_{B}$ is expressed as follows

(y - y_{A}) (x_{A} - x_{B}) - (y_{A} - y_{B}) (x - x_{A}) = 0

Use the new information to solve the tasks.

Task 2

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nagranie abstraktu

Rewrite the equation of linelineline $k : y + \frac{x - 3}{2} = \frac{x + 1}{5}$ in the standard form.

$3 x + 10 y - 17 = 0$

Task 3

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nagranie abstraktu

Find the standard equation of linelineline $k$ , on which lie two points $A (- 3; - 4)$ and $B (4; 7)$ .

$7 x - 11 y + 49 = 0$

Task 4

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Source: GroMar, licencja: CC BY 3.0.

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nagranie abstraktu

Write the standard equation of a lineequation of a lineequation of a line intersecting at two points $A (2; 6)$ and $B (4; 8)$ . Check if point $C (- 1; - 3)$ lies on this linelineline.

$x - y + 4 = 0$ , doesn’t lie on the line

Task 5

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Source: GroMar, licencja: CC BY 3.0.

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nagranie abstraktu

Point $A (a; 3)$ lies on linelineline $2 x + y - 5 = 0$ . Calculate $a$ .

$a = 1$

Task 6

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nagranie abstraktu

An extra task:

For what value of number $m$ lines $k : x - m y + 4 + m = 0$ and $l : 2 m x + y - m - 1 = 0$ intersect on the OY axis?

$m \in {- 2, 2}$

Having finished all the tasks, do the consolidation tasks. Formulate the conclusion to memorize.

A linelineline perpendicular to OX axis cannot be described with the slope‑intercept equation.
In order to convert from the slope‑intercept equation of a lineequation of a lineequation of a line to the standard equation, you need to move all terms of the equation onto one side and multiply both sides of the equation by the same number, so that all the obtained coefficients are integers.
In order to convert from the standard equation of a lineequation of a lineequation of a line $A x + B y + C = 0$ to the slope‑intercept equation, you need to move terms $A_{x}$ and $C$ onto the right side of the equation. If $B \neq 0$ , you need to divide both sides of the equation by $B$ .
The standard equation of a lineequation of a lineequation of a line intercepting at two points $A (x_{A}; y_{A})$ and $B (x_{B}; y_{B})$ , where $x_{A} \neq x_{B}$ is expressed as follows $(y - y_{A}) (x_{A} - x_{B}) - (y_{A} - y_{B}) (x - x_{A}) = 0$ .

Exercises

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Exercise 1

Point P (6; 3) lies line expressed with 3x + my + m + 2 = 0. So:

m = -6
m = -5
m = -4
m = -3

Exercise 2

Draw line 3x+y‑6=0.

Find the coordinates of intersection points of the line with the axes of the coordinate system.
Calculate the area of the triangle bounded by this Line and the axes of the coordinate system.

$(0; 6)$ and $(2; 0)$
$6$

Exercise 3

Describe in English how to convert from the slope‑intercept equation of a line to the standard equation.

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Exercise 4

Indicate which pairs of expressions or words are translated correctly.

prosta - line
równanie prostej - equation of a line
równanie prostej w postaci kierunkowej - slope-intercept equation of a line
równanie prostej w postaci ogólnej - standard equation of a line
współczynniki liczbowe - coordinate system
układ współrzędnych - cooeficients

zadanie

Source: GroMar, licencja: CC BY 3.0.

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Match Polish terms with their English equivalents.

równanie prostej w postaci kierunkowej
standard equation of a line
equation of a line
cooeficients
slope-intercept equation of a line
prosta
równanie prostej w postaci ogólnej
równanie prostej
Współczynniki liczbowe
line

Source: Zespół autorski Politechniki Łódzkiej, licencja: CC BY 3.0.

Glossary

cooeficients

współczynniki liczbowe

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wymowa w języku angielskim: cooeficients

coordinate system

układ współrzędnych

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Source: GroMar, licencja: CC BY 3.0.

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wymowa w języku angielskim: coordinate system

equation of a line

równanie prostej

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wymowa w języku angielskim: equation of a line

line

prosta

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wymowa w języku angielskim: line

slope‑intercept equation of a line

równanie prostej w postaci kierunkowej

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Source: GroMar, licencja: CC BY 3.0.

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wymowa w języku angielskim: slope‑intercept equation of a line

standard equation of a line

równanie prostej w postaci ogólnej

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Source: GroMar, licencja: CC BY 3.0.

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wymowa w języku angielskim: standard equation of a line

y‑axis

oś rzędnych

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wymowa w języku angielskim: y‑axis

Keywords

cooeficientscooeficientscooeficients

equation of a lineequation of a lineequation of a line

linelineline

slope‑intercept equation of a lineslope‑intercept equation of a lineslope‑intercept equation of a line

standard equation of a linestandard equation of a linestandard equation of a line

Scenariusz

Exercises

Glossary

Keywords