The Straight line and its slope

What is a Straight Line?

A straight line is a locus of all points such that the moving point does not change its direction, or the rate of change of direction remains constant. In other words, a straight line is a line of zero curvature. Precisely, a straight line is the shortest distance between two points in a plane.

The general form of a straight line in two-dimensional Cartesian coordinates is:

𝑦 = 𝑚𝑥 + 𝑐

Where:

𝑚 = the slope or gradient of the line.

𝑐 = the y-intercept, i.e., the value of 𝑦 where the graph of the line crosses the y-axis.

What is Curvature?

Curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. A straight line has zero curvature, which means it does not bend.

Plotting a Straight Line

Let us take a simple linear equation:

$$\mathrm{<b\; style="text-align:\; center;"><span\; style="color:\; \#0c343d;">𝑦\; =\; 2𝑥\; +\; 1</span></b>}$$

Computing the values of $y$ for various $x$:

• When $x = 0$:

  $$y=2(0)+1=1$$

• When $x=\mathrm{1:}$

  $$y=2(1)+1=3$$

• When $x = 2$:

  $$y=2(2)+1=5$$

• When $x = 3$:

  $$y=2(3)+1=7$$

• When $x = 4$:

  $$y = 2(4) + 1 = 9$$

Solutions:

x	0	1	2	3	4
y=2x+1	1	3	5	7	9

y=2x+1 | Desmos

Locating the Position of a Straight Line in the Plane

We can determine the position of a straight line on a plane by:

• Any two points through which the straight line passes.

• Y-intercept and the angle it makes with the X-axis.

• Intercepts on both X and Y axes

​Straight Line Formulae 

Slope (Gradient) of a Straight Line given Two Points

Given two points $A(x_1, y_1)$ and $B(x_2, y_2)$:

$$\boxed{{\displaystyle \mathrm{<span\; style="color:\; \#0c343d;">m</span>}<span\; style="color:\; \#0c343d;">=</span>\frac{{\mathrm{<span\; style="color:\; \#0c343d;">y</span>}}_{\mathrm{<span\; style="color:\; \#0c343d;">2</span>}}<span\; style="color:\; \#0c343d;">-</span>{\mathrm{<span\; style="color:\; \#0c343d;">y</span>}}_{\mathrm{<span\; style="color:\; \#0c343d;">1</span>}}}{{\mathrm{<span\; style="color:\; \#0c343d;">x</span>}}_{\mathrm{<span\; style="color:\; \#0c343d;">2</span>}}<span\; style="color:\; \#0c343d;">-</span>{\mathrm{<span\; style="color:\; \#0c343d;">x</span>}}_{\mathrm{<span\; style="color:\; \#0c343d;">1</span>}}}}}$$

Equation of a Straight Line

a) Slope-Intercept Form:

$$\boxed{{\displaystyle \mathrm{<b><span\; style="color:\; \#0c343d;">y</span></b>}<b><span\; style="color:\; \#0c343d;">=</span></b>\mathrm{<b><span\; style="color:\; \#0c343d;">m</span></b>}\mathrm{<b><span\; style="color:\; \#0c343d;">x</span></b>}<b><span\; style="color:\; \#0c343d;">+</span></b>\mathrm{<b><span\; style="color:\; \#0c343d;">c</span></b>}}}$$

• $m$: slope

• $c$: y-intercept (value of $y$ when $x = 0$)

b) Point-Slope Form:

$$\boxed{{\displaystyle \mathrm{<span\; style="color:\; \#0c343d;"><b>y</b></span>}<span\; style="color:\; \#0c343d;"><b>-</b></span>{\mathrm{<span\; style="color:\; \#0c343d;"><b>y</b></span>}}_{\mathrm{<span\; style="color:\; \#0c343d;"><b>1</b></span>}}<span\; style="color:\; \#0c343d;"><b>=</b></span>\mathrm{<span\; style="color:\; \#0c343d;"><b>m</b></span>}<span\; style="color:\; \#0c343d;"><b>(</b></span>\mathrm{<span\; style="color:\; \#0c343d;"><b>x</b></span>}<span\; style="color:\; \#0c343d;"><b>-</b></span>{\mathrm{<span\; style="color:\; \#0c343d;"><b>x</b></span>}}_{\mathrm{<span\; style="color:\; \#0c343d;"><b>1</b></span>}}<span\; style="color:\; \#0c343d;"><b>)</b></span>}}$$

• $<span\; style="color:\; \#0c343d;">(</span>{\mathrm{<span\; style="color:\; \#0c343d;">x</span>}}_{\mathrm{<span\; style="color:\; \#0c343d;">1</span>}}<span\; style="color:\; \#0c343d;">,</span>{\mathrm{<span\; style="color:\; \#0c343d;">y</span>}}_{\mathrm{<span\; style="color:\; \#0c343d;">1</span>}}<span\; style="color:\; \#0c343d;">):\; a\; known\; point</span>$

• $m$: slope

c) Two-Point Form (when two points are known):

$$\boxed{{\displaystyle \mathrm{<b><span\; style="color:\; \#0c343d;">y</span></b>}<b><span\; style="color:\; \#0c343d;">-</span></b>{\mathrm{<b><span\; style="color:\; \#0c343d;">y</span></b>}}_{\mathrm{<b><span\; style="color:\; \#0c343d;">1</span></b>}}<b><span\; style="color:\; \#0c343d;">=</span></b>\frac{{\mathrm{<b><span\; style="color:\; \#0c343d;">y</span></b>}}_{\mathrm{<b><span\; style="color:\; \#0c343d;">2</span></b>}}<b><span\; style="color:\; \#0c343d;">-</span></b>{\mathrm{<b><span\; style="color:\; \#0c343d;">y</span></b>}}_{\mathrm{<b><span\; style="color:\; \#0c343d;">1</span></b>}}}{{\mathrm{<b><span\; style="color:\; \#0c343d;">x</span></b>}}_{\mathrm{<b><span\; style="color:\; \#0c343d;">2</span></b>}}<b><span\; style="color:\; \#0c343d;">-</span></b>{\mathrm{<b><span\; style="color:\; \#0c343d;">x</span></b>}}_{\mathrm{<b><span\; style="color:\; \#0c343d;">1</span></b>}}}<b><span\; style="color:\; \#0c343d;">(</span></b>\mathrm{<b><span\; style="color:\; \#0c343d;">x</span></b>}<b><span\; style="color:\; \#0c343d;">-</span></b>{\mathrm{<b><span\; style="color:\; \#0c343d;">x</span></b>}}_{\mathrm{<b><span\; style="color:\; \#0c343d;">1</span></b>}}<b><span\; style="color:\; \#0c343d;">)</span></b>}}$$

d) Intercept Form:

$$\boxed{{\displaystyle \frac{\mathrm{<b><span\; style="color:\; \#0c343d;">x</span></b>}}{\mathrm{<b><span\; style="color:\; \#0c343d;">a</span></b>}}<b><span\; style="color:\; \#0c343d;">+</span></b>\frac{\mathrm{<b><span\; style="color:\; \#0c343d;">y</span></b>}}{\mathrm{<b><span\; style="color:\; \#0c343d;">b</span></b>}}<b><span\; style="color:\; \#0c343d;">=</span></b>\mathrm{<b><span\; style="color:\; \#0c343d;">1</span></b><span\; style="color:\; \#0c343d;"></span>}}}$$

• $a$: x-intercept

• $b$: y-intercept

Condition for Parallel Lines