Question

Another line “r” is perpendicular to the line 3y = 6x + 9. Find the gradient of the line “r”.

Answer 1

We are given a line:

$3y =6x+9$

Please refer to the attached graph for having a look at the graph of the given equation.

Let us convert the given equation to simplified form:

Dividing the given equation  $3y = 6x + 9$ with 3 on both sides:

$\Rightarrow \dfrac{3y}{3}= \dfrac{6x + 9}{3}\\\Rightarrow y= 2x + 3$

The given equation is now converted to $y=2x+3$ ...... (1)

The slope equation of a line is given as:

$y=mx+c$ ..... (2)

where (x, y) are coordinates of points on line.

m is the gradient/slope of the line

c is the y intercept.

Let slope of given line be $m_1$

Comparing the equations (1) and (2):

We get:

Let Slope/gradient of given line, $m_1$ = 2

Let the gradient/slope of the line 'r' which is perpendicular to the given line = $m_2$

As per the property of gradients/slopes of perpendicular lines:

The product of gradient/slope of perpendicular lines is equal to -1.

i.e. $m_1$ $\times$ $m_2$ = -1

$\Rightarrow 2 \times m_2=-1\\\Rightarrow m_2=-\dfrac{1}{2}$

So, the gradient/slope of line 'r' perpendicular to the given line is:

$\dfrac{1}2$