Question

What is an equation of the line in the form $ax + by + c = 0$ with gradient $- 2$ through the point $(4,\ - 6)$ ?

Answer 1

In this question, we need to find the equation of the line which is in the form of $ax+by+c=0$ . Also the gradient of the line is $-2$ . The gradient is the same as the slope. Moreover gradients can have a positive as well as negative value. Graphically , The gradient of a horizontal line is zero and also the gradient of the $x$ axis that is a horizontal line is zero. And also the line passes through the line $(4,-6)$ . Here we need to find the equation of the line.
Formula used :
Gradient or Slope ,
$m\ = \dfrac{\left({change\ in\ y} \right)}{{change\ in\ x}}$
$m = \dfrac{\left(y{_2}- y{_1}\right)}{\left(x{_2} – x{_1}\right)}$

Complete step-by-step solution:
Given, gradient is $- 2$ and the point $(x{_1}, x{_2})$ is $(4,\ - 6)$
$\Rightarrow \ m = - 2$
We know the formula of gradient, $m = \dfrac{\left(y{_2}- y{_1}\right)}{\left(x{_2} – x{_1}\right)}$
Thus, $- 2 = \dfrac{\left(y{_2}- y{_1}\right)}{\left(x{_2} – x{_1}\right)}$
$\Rightarrow \ - 2 = \dfrac{\left( y - \left( - 6 \right) \right)}{x – 4}$
By cross multiplying,
We get,
$- 2(x – 4)\ = (y – ( - 6))$
On simplifying,
We get,
$\Rightarrow -2x+8 = y+6$
By moving all the terms to one side,
We get,
$\Rightarrow - 2x-y+8–6 = 0$
On simplifying,
We get,
$\Rightarrow -2x – y + 2 = 0$
$\Rightarrow 2x+y-2 = 0$
Thus the equation of line is $2x+y-2 = 0\$ which is in the form of $ax + by + c = 0$
Final answer :
The equation of line is $2x + y - 2 = 0\$ which is in the form of $ax + by + c = 0$

Note: The slope of a line is defined as the measure of its Steepness. It is calculated by dividing the change in $y$ coordinate by change in $x$ co-ordinate. Mathematically, slope is denoted by the letter $m$. Slope is positive when m is greater than $0$ and when m is less than $0$ , slope is negative. If the slope is equal to $0$ That means it is a constant function. Graphically, The gradient of two parallel lines is equal and also the product of the gradients of two perpendicular lines is $- 1$.