Straight lines – Equation – Slope – Solved Examples

Straight lines

A straight line is a combination of many points along a straight path having no end in both directions. 

It doesn’t have a definite length and has zero width.

Every straight line can be expressed in the form of a linear equation (ax + by = c) and if there is any point satisfying that equation, then it lies on that line.

Every point on the straight line have two parts:

1. Absicca (also known as the x-coordinate)
2. Ordinate (also known as the y-coordinate)

The inclination of a straight line

The angle between the line and the positive x-axis is known as the inclination of the line. It can be positive or negative based on the direction of measurement. 

Important Points

• The inclination of the x-axis or any line parallel to the x-axis is 0°.
• The inclination of the y-axis or any line parallel to the y-axis is 90°.

The slope of a line

The slope of a line is defined as the measure of the steepness of the same line. It is the tangent of the inclination angle of that line. It can be also referred to as the ratio of the difference between the ordinates to the difference between the abscissa of any two points on the line.

\text { Slope }=\frac{\text { Difference between ordinates (y coordinates) }}{\text { Difference between abscissa ( } x \text { coordinates) }}

Example

A line is passing through points point P and Q having coordinates \left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right)respectively 

The slope of \mathrm{PQ}=\mathrm{m}=\tan \theta=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}

Important Points

• The slope is (+ve) when the line makes an acute angle with the positive x-axis. 
• The slope is (-ve) when the line makes an obtuse angle with the positive x-axis.
• When the slope is equal to 0, it implies the line is parallel to the x-axis.
• The slope of the y-axis or any line parallel to the y-axis is not defined.

Parallel lines

When a line is parallel to another line, then their slopes are equal to each other 

m_{1}=m_{2}

Perpendicular lines

When a line is perpendicular to another line, the product of their respective slopes is equal to (-1).

Let there be a line containing points P\left(x_{1}, y_{1}\right)\text{ and } Q\left(x_{2}, y_{2}\right)and the other line contains points A\left(x_{3}, y_{3}\right)\text{ and } B\left(x_{4}, y_{4}\right).

If PQ is perpendicular to AB:

\Rightarrow \mathrm{m}_{1} \times \mathrm{m}_{2}=(-1)
\Rightarrow \frac{y_{1}-y_{2}}{x_{1}-x_{2}} \times \frac{y_{3}-y_{4}}{x_{3}-x_{4}}=(-1)

Collinearity of three points

Let there be 3 points P\left(x_{1}, y_{1}\right), Q\left(x_{2}, y_{2}\right), R\left(x_{3}, y_{3}\right)

P, Q and R will be collinear when the slope of the line segment PQ will be equal to the slope of the line segment QR.

\frac{y_{1}-y_{2}}{x_{1}-x_{2}}=\frac{y_{2}-y_{3}}{x_{2}-x_{3}}

X- Intercepts & Y- Intercepts

When a line intersects with the x-axis, the distance between the origin and the point of intersection is known as the x-intercept.

When a line intersects with the y-axis, the distance between the origin and the point of intersection is known as the y-intercept.

Equation of a straight line 

• Point – Slope form

A line is passing through a point having coordinates \left(x_{1}, y_{1}\right) and its slope is ‘m’

Equation: 

y-y_{1}=m\left(x-x_{1}\right)

• Slope – Intercept form

A line having y-intercept equal to ‘c’ and slope equal to “m”

Equation:

y=m x+c

• Two-point form

A line is passing through two points having coordinates \left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right)

Equation:

y-y_{1}=\frac{y_{1}-y_{2}}{x_{1}-x_{2}}\left(x-x_{1}\right)

• Intercept form

A line having x-intercept and y-intercept equal to ‘a’ and ‘b’ respectively.

\frac{x}{a}+\frac{y}{b}=1

This line also passes through the point (0,b) and (a,0).

Solved Examples

1. The x coordinate of a point on the line (y = 2x + 3) is 3. Find its y coordinate.

We know that every point present on the line satisfies its equation.

x coordinate of the point = 3

For finding y-coordinate, we have to put the value of x in the given equation

Equation of the line
y = 2x + 3

⇒ y = 23 + 3 = 6 + 3 = 9

Hence the y coordinate of the point is 9.

2. A straight line is having a slope equal to 2. Find the slope of the line perpendicular to the given line.

When a line is perpendicular to another line, the product of their respective slopes is equal to (-1).

\Rightarrow m_{1} \times m_{2}=(-1)
\Rightarrow 2 \times m_{2}=(-1)
\Rightarrow m_{2}=(-1 / 2)

3. The equation of a line BC is 4x + 7 = 9. Find the value of its slope (m) and y-intercept (c)

We have to write the given equation in slope-intercept form to find the value of ‘m’ and ‘c’.

      4 x+7=9
\Rightarrow  \frac{4 x}{4}+\frac{7}{4}=\frac{8}{4}
\Rightarrow  x+\frac{7}{4}=\frac{9}{4}
\Rightarrow  m=\frac{7}{4} \text { and } c=\frac{9}{4}

\text { Hence the slope is equal to } \frac{7}{4} \text { and the } y \text {-intercept is equal to } \frac{9}{4}.