Equation of a line in the standard form and in the slope-intercept form

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Równanie prostej w postaci ogólnej oraz w postaci kierunkowej

Learning objectives

You will learn to find the equation of a line passing through two different points in the slope‑intercept form if it is possible and the standard form.

Learning effect

You can find the equation of a line passing through two different points in the slope‑intercept form and the standard form.
You can describe in English how to find the equation of a line passing through two points.

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nagranie abstraktu

The linear function formula: $f (x) = a x + b$ .
Formula $a = \frac{y_{B} - y_{A}}{x_{B} - x_{A}}$ for the slopeslopeslope of a line passing through two different points $A = (x_{A}, y_{A})$ and $B = (x_{B}, y_{B})$ so than $x_{A} \neq x_{B}$ .
Ggeometrical interpretation of coefficient $b$ in the formula for a linear function.

Task 1

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nagranie abstraktu

Function $f$ determined by any real number x with formula $f (x) = 3 x + 2$ is given.

a, What is this type of a function called?
b. What is the graph of this function?
c. What are coefficients 3 and 2 called in the formula for function $f$ ?
d. What is the geometric sense of coefficients 3 and 2?
e. Which diagram presents this function?

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Rysunek przedstawia cztery układy współrzędnych oznaczone literami od A do D. W każdym z układów narysowana jest prosta. W punkcie A prosta przechodzi przez punkty o współrzędnych (minus dwa, cztery) i (jeden, trzy), w punkcie B prosta przechodzi przez punkty o współrzędnych (minus jeden, minus jeden) i (dwa, zero), w punkcie C prosta przechodzi przez punkty o współrzędnych (minus dwa, minus jeden) i (dwa, minus jeden), w punkcie D prosta przechodzi przez punkty o współrzędnych (zero, zero) i (pięć, minus jeden). — Source: GroMar, licencja: CC BY 3.0.

Task 2

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nagranie abstraktu

Write down the equation of a line presented in diagrams A – D in Task 1 in the slope‑intercept formslope‑intercept formslope‑intercept form.

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nagranie abstraktu

Consider the following example. The equation of a line presented in the diagram $y = 0 \cdot x + 3$ in the slope‑intercept form.

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Rysunek przedstawia prostą w układzie współrzędnych, przechodząca przez punkty o współrzędnych (minus trzy, trzy) oraz (pięć, trzy). — Source: GroMar, licencja: CC BY 3.0.

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nagranie abstraktu

How to describe any point located on this line with coordinates?
How to write down the equation of a line passing through points $A = (4, - 1)$ and $B = (4, 5)$ ?
Which other points are located on line $A B$ ?

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nagranie abstraktu

Any line on the Cartesian plane can be described with equation $A x + B y + C = 0$ where coefficients $A$ and $B$ cannot equal 0 at the same time. Yet, if $A = 0$ , the line is parallel to X‑axis and if $B = 0$ , the line is parallel to Y‑axis.

The equation of a line in the standard form

Definition: The equation of a line in the standard form

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nagranie abstraktu

Equation $A x + B y + C = 0$ when $A$ and $B$ don’t equal 0 at the same time is called the equation of a line in the standard formstandard form standard form.

Solve the task:

Task 3

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nagranie abstraktu

Triangle $ABC$ , in which $A = (- 3, 1)$ , $B = (3, 5)$ , $C = (3, 1)$ is given. Write down the equations of lines including sides $AB$ and $AC$ of triangle $ABC$ . Use formula $y = \frac{y_{B} - y_{A}}{x_{B} - x_{A}} (x - x_{A}) + y_{A}$ for the equation of a line passing through two different points $A = (x_{A}, y_{A})$ and $B = (x_{B}, y_{B})$ so that $x_{A} \neq x_{B}$ .

Think:

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nagranie abstraktu

Can the presented formula be used to write down the equation of a line including side $BC$ of triangle $ABC$ ? If not, why?
How to convert the equation so that it can be used to find the equation of line $BC$ ?

It isn’t possible. The reason is the equality of X‑axis for points $B$ and $C$ so $x_{B} = x_{C}$ , because the denominator of the fraction $\frac{y_{C} - y_{B}}{x_{C} - x_{B}}$ will equal 0.
Multiply both sides of the equation $y = \frac{y_{B} - y_{A}}{x_{B} - x_{A}} (x - x_{A}) + y_{A}$ by $x_{B} - x_{A}$ and write it down in the equivalent form: $(y_{B} - y_{A}) (x - x_{A}) - (x_{B} - x_{A}) (y - y_{A}) = 0$ .

Any line passing through two different points $A = (x_{A}, y_{A})$ and $B = (x_{B}, y_{B})$ can be described with this equation, which is called the standard equation in the vertex form.

Task 4

Using the applet practice finding the equations for set points $A$ and $B$ .

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Rysunek przedstawia układ współrzędnych oraz prostą przechodzącą przez dwa punkty: A o współrzędnych trzy, trzy oraz B o współrzędnych minus cztery, zero. — Source: GroMar, licencja: CC BY 3.0.

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Task 5

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nagranie abstraktu

Open Geogebra applet „Equations of a line passing through two set points”. Change the position of points $A$ and $B$ . Find the slopeslopeslope of line $AB$ and the equation of this line in the slope‑intercept and the standard formstandard form standard form.

Check your results.

Task 6

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nagranie abstraktu

The standard equation of a line is given. Convert it into the slope‑intercept formslope‑intercept formslope‑intercept form if possible.

a. $2 x - 3 y + 9 = 0$
b. $- x + 2 y + 4 = 0$
c. $2 x + 3 y = 0$
d. $x - 0 \cdot y + 9 = 0$

Task 7

An extra task:

Prove that any line with the equation $(m^{2} + 1) x + (m^{2} - 1) y - 2 = 0$ intercepts point $M = (1, - 1)$ , but none of the lines intercepts point $N = (- 1, - 1)$ .

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nagranie abstraktu

The slope‑intercept equation of a line: $y = a x + b$ .
The standard equation of a line: $A x + B y + C = 0$ , where coefficients $A$ and $B$ cannot equal 0 at the same time. Yet, if $A = 0$ , the line is parallel to X‑axis and if $B = 0$ , the line is parallel to Y‑axis.
The equation of a line passing through two different points:
$(y_{B} - y_{A}) (x - x_{A}) - (x_{B} - x_{A}) (y - y_{A}) = 0$
so that $A = (x_{A}, y_{A})$ and $B = (x_{B}, y_{B})$ .

Exercises

Exercise 1

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Wersja alternatywna ćwiczenia: Determine which sentences are true. Możliwe odpowiedzi: 1. The slope of a line passing through the origin of the coordinate system and point

A = (2, - 3)

equals

\frac{3}{2}

., 2. The line passing through points

A = (0, 5)

and

B = (2, - 3)

can be described with the slope-intercept equation

y = - 4 x + 5

., 3. The line passing through points

A = (- 21, 2)

and

B = (21, 2)

cannot be described with the slope-intercept equation., 4. The line passing through points

A = (\frac{1}{4}, - \frac{3}{2})

and

B = (3, - 2)

can be described with equation

y = - \frac{2}{11} (x - 3) - 2 .

Determine which sentences are true.

The slope of a line passing through the origin of the coordinate system and point $A = (2, - 3)$ equals $\frac{3}{2}$ .
The line passing through points $A = (0, 5)$ and $B = (2, - 3)$ can be described with the slope-intercept equation $y = - 4 x + 5$ .
The line passing through points $A = (- 21, 2)$ and $B = (21, 2)$ cannot be described with the slope-intercept equation.
The line passing through points $A = (\frac{1}{4}, - \frac{3}{2})$ and $B = (3, - 2)$ can be described with equation $y = - \frac{2}{11} (x - 3) - 2 .$

zadanie

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Exercise 2

Points: $A = (- 2, - 1)$ , $B = (1, - 2)$ , $C = (7, 1)$ , $D = (1, 5)$ are given. Find the equations of lines including the sides and the diagonals of quadrangle $ABCD$ . Write down each of the equations in the standard form and in the slope‑intercept form if possible.

Line $AB$ : $y = - \frac{1}{3} x - \frac{5}{3}$ (slope‑intercept), $x + 3 y + 5 = 0$ (standard)

Line $BC$ : $y = \frac{1}{2} x - \frac{5}{2}$ (slope‑intercept), $x - 2 y - 5 = 0$ (standard)

Line $CD$ : $y = - \frac{2}{3} x + \frac{17}{3}$ (slope‑intercept), $2 x + 3 y - 17 = 0$ (standard)

Line $AD$ : $y = 2 x + 3$ (slope‑intercept), $2 x - y + 3 = 0$ (standard)

Line $AC$ : $y = \frac{2}{9} x - \frac{5}{9}$ (slope‑intercept), $2 x - 9 y - 5 = 0$ (standard)

Line $BD$ : there is no slope‑intercept, $x - 1 = 0$ (standard)

Exercise 3

Points $A = (4, 10)$ and $B = (6, 12)$ are given. Find the slope‑intercept equation of the line passing through these points. Write down your line of reasoning in English.

At first I find the slope of this line using the formula $a = \frac{y_{B} - y_{A}}{x_{B} - x_{A}}$ .

so $a = \frac{12 - 10}{6 - 4} = 1$ .

Next, I choose one of the points, for example point $A$ and put its coordinates and the calculated before slope a into equation of a line $y = a (x - x_{A}) + y_{A}$ .

In this way I get the equation of the line passing through points A and B:

$y = 1 \cdot (x - 4) + 10$ , thus $y = x + 6$ .

Exercise 4

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Indicate which pairs of expressions or words are translated correctly.

współczynnik kierunkowy - slope
równanie kierunkowe prostej - point-slope form
wyraz wolny w równaniu prostej - y-intercept
równanie ogólne prostej - two-point form
równanie prostej w postaci kierunkowej - standard form

zadanie

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Match Polish terms with their English equivalents.

współczynnik kierunkowy
y-intercept
slope
slope-intercept form
postać kierunkowa prostej
wyraz wolny w równaniu prostej
postać ogólna prostej
standard form

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

Glossary

point‑slope form

równanie kierunkowe prostej

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wymowa w języku angielskim: point‑slope form

slope

współczynnik kierunkowy

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wymowa w języku angielskim: slope

slope‑intercept form

równanie kierunkowe prostej

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wymowa w języku angielskim: slope‑intercept form

standard form

równanie ogólne prostej

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wymowa w języku angielskim: standard form

two‑point form

równanie prostej w postaci kierunkowej

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wymowa w języku angielskim: two‑point form

y‑intercept

wyraz wolny w równaniu prostej

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wymowa w języku angielskim: y‑intercept

Keywords

slopeslopeslope

slope‑intercept formslope‑intercept formslope‑intercept form

standard formstandard form standard form

y‑intercepty‑intercepty‑intercept

Scenariusz

Exercises

Glossary

Keywords